Paper Title: Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability.



Grothendieck Shenanigans]Grothendieck Shenanigans:

Permutons from pipe dreams via integrable probability





[
    
	A. H. Morales,
	G. Panova,
	L. Petrov,
	D. Yeliussizov
    
============================================================



§ ABSTRACT 
	We study random permutations arising from reduced pipe dreams. Our
	main model is motivated by  Grothendieck
	polynomials with parameter β=1 arising in
	K-theory of the flag variety. The probability weight of a
	permutation is proportional to the principal specialization
	(setting all variables to 1)
	of the
	corresponding
	Grothendieck polynomial. By mapping
	this random permutation to a version of TASEP (Totally
	Asymmetric Simple Exclusion Process), we describe the
	limiting permuton and fluctuations around it as the order
	n of the permutation grows to infinity. The fluctuations
	are of order n^1/3 and have the Tracy–Widom GUE distribution,
	which places this algebraic (K-theoretic) model
	into the Kardar–Parisi–Zhang universality class.

	We also investigate non-reduced pipe dreams and make
	progress on a recent open problem on the asymptotic number
	of inversions of the resulting permutation. Inspired by
	Stanley's question for the maximal value of principal
	specializations of Schubert polynomials, we resolve the
	analogous question for β=1 Grothendieck polynomials,
	and provide bounds for general β.




§ INTRODUCTION





 §.§ Overview



Algebraic Combinatorics established itself as a field that uses combinatorial methods to understand algebraic behavior in problems ranging from Group Theory to Algebraic Geometry. It started with stark exact formulas, like the celebrated hook-length formula for the dimension of irreducible modules of the symmetric group S_n; beautiful interpretations, like the Littlewood–Richardson rule for the structure constants of representations of the general linear group GL_n; powerful and intricate bijections, like the Robinson–Schensted–Knuth correspondence. However, exact answers only go so far, leaving room for questions like  “approximately how many”, “what are the typical objects”,  and “what is the typical behavior”. These questions lead us into the realm of Asymptotic Algebraic Combinatorics, which aims to answer them with the help of tools originating outside of Combinatorics. In the present work, we employ Integrable Probability, a rapidly evolving field focused on developing and analyzing interacting particle systems and random growth models possessing a certain degree of structure or symmetry.

Many significant questions concerning exact formulas and combinatorial interpretations in cohomology and K-theory of the flag variety remain open. Understanding their asymptotic behavior is thus even more natural. Stanley <cit.> asked the most basic question on the  principal specializations of Schubert polynomials 𝔖_w
(representing cohomology classes of the flag variety): does the following limit exist

    lim_n→∞1/n^2log_2 max_w ∈ S_n𝔖_w(
    	1,…,1 _n),

and if so, what is it and for which permutations w is this achieved.
This question (including the existence of the limit) is still open.
In <cit.>, a lower bound of about 0.29 was established for layered permutations. An upper bound of about 0.37 comes from a remarkable connection with Alternating Sign Matrices and the six-vertex model (see <Ref>).

A natural one-parameter generalization of Schubert
polynomials are the Grothendieck polynomials
𝔊_w^β, which represent K-theoretic
classes of the flag variety. Extending Stanley's questions, we
would like to understand the asymptotic behavior of maximal
 principal specializations max_w∈ S_n𝔊_w^β(1^n)
of Grothendieck polynomials. This
question was first touched on in
<cit.>, <cit.>. Establishing a family of
such maximal permutations would shed light on the
Schubert questions as β→ 0.

Thanks to a combinatorial model for Grothendieck and
Schubert polynomials, these questions have very natural
statistical mechanics interpretations.
Namely, both polynomials are partition functions of tilings
into crosses and elbows of a size n triangle (staircase) shape,
which result in a configuration of n “pipes”. Such pipe configurations
are often called pipe dreams.
In the Schubert case, the only valid
tilings are the ones where no two pipes cross more than
once. This is a global (long-range interaction) condition,
which makes the model much less tractable.
In the Grothendieck case
for β=1, all tilings are allowed, but the pipes must be
resolved (reduced) to obtain a permutation. This model can be mapped to a
colored stochastic six-vertex model
(and furter to TASEP, the Totally Asymmetric Simple Exclusion Process).
These interacting particle systems have only local (short-range) interactions, and are
amenable to techniques from Integrable Probability.



In this work,
we investigate the asymptotics of
β=1 Grothendieck random permutations
and characterize their limit shape which is described by a permuton.
In principle, the typical
permutation (sampled from the limiting permuton) should give the
asymptotically maximal principal specialization
𝔊_w^β=1(1^n), but we do not address this question here.
We also study fluctuations of Grothendieck random permutations around the limiting permuton.
They
are of order n^1/3 and asymptotically have the Tracy–Widom GUE distribution.
This distribution was
first observed in the fluctuations of
the largest eigenvalue of Gaussian
random matrices
with unitary symmetry. By now, having Tracy–Widom fluctuations is an indication that a
model is within the
Kardar–Parisi–Zhang (KPZ) universality class <cit.>, which includes a wide range of
random growth models and interacting particle systems.



We also study the natural, yet not as algebraically
motivated, model, where the staircase is tiled with crosses
with probability p and elbows with probability 1-p, and
the pipes follow the tiles without any reductions or
reassignments. Using simple random walks, we
show that the resulting random permutation is close to the identity, and, moreover,
that the total
displacement
∑_i=1^n|i-w_i|
and length (number of inversions)
of this permutation
are of order n^3/2. This partially resolves a recent open problem
posed by
Colin Defant.



Returning to the original question, we also investigate the asymptotics
of max_w ∈ S_n𝔊_w^β(1^n).
When
β=1, this maximum is known to be O(2^n2). Using the
correspondence with
2-enumerated Alternating Sign Matrices
(equivalently, the six-vertex model with domain wall boundary conditions
and free-fermion weights; or the model of uniform domino tilings of the Aztec diamond),
we show that a large family of layered
permutations also achieve this asymptotic maximum. For
general β, we establish certain bounds for the maximal
 principal specialization.



 §.§ Grothendieck polynomials



The nonsymmetric Grothendieck polynomials generalize the Schubert and the Schur polynomials,
and capture the K-theory of the flag variety.
They can be defined or interpreted in a number of ways,
including via
divided difference operators <cit.>,
pipe dreams <cit.>, bumpless pipe dreams
<cit.>,
<cit.>,
and solvable lattice models
<cit.>, <cit.>.

We denote by S_n the set permutations of
{1,2,…,n} that we write in the one-line notation
w=w_1w_2⋯ w_n unless indicated otherwise.
We also denote the image of i under w by w(i), and use the notation
w_i=w(i) interchangeably when this does not lead to confusion.
The longest
permutation n n-1… 2 1 is denoted by w_0=w_0(n).
For a permutation w of length ℓ, we denote by R(w)
the set of reduced words of w, that is, tuples
(r_1,…,r_ℓ) such that s_r_1⋯ s_r_ℓ
is a reduced decomposition of w, where s_i=(i, i+1) are the
simple transpositions.

The (single) Grothendieck polynomials were introduced
by Lascoux <cit.> to study the K-theory of
flag varieties. Their original recursive definition is as
follows. Let π_i:ℤ[β][ x]→ℤ[β][ x] denote the isobaric
divided difference operator:

    π_i f(1+β x_i+1) f-(1+β x_i) s_i·  f/x_i-x_i+1,

where s_i acts on a polynomial f by permuting x_i and x_i+1.
The Grothendieck polynomials 𝔊_w^β are recursively
determined by the following conditions:

	
  * For the longest permutation, we have
		𝔊_w_0(n)^β=(1+β)^n2
		x_1^n-1x_2^n-2⋯ x_n-1.
	
  * For all w∈ S_n and i=1,…, n-1
		such that
		ℓ(w s_i)=ℓ(w)+1,
		we have
		𝔊_w^β =
		π_i 𝔊_ws_i^β.

Setting β=0 in 𝔊_w^β, we obtain a Schubert
polynomial 𝔖_w which represents cohomology
classes of Schubert cycles in the flag variety <cit.>.



 §.§ Pipe dreams and permutations



Grothendieck polynomials can also be defined combinatorially
via pipe dreams (equivalently, rc-graphs).

A pipe dream
of order n
is a tiling of the
staircase shape
(having n-1 boxes in the first row,
n-2 boxes in the second row, and so on,
with boxes left-justified)
by tiles of two types:
bumps
-2pt
    < g r a p h i c s >

and
crossings
-2pt
    < g r a p h i c s >
.
The n-th diagonal below the staircase is
equipped with half bumps
-2pt
    < g r a p h i c s >

(whose boundary we do not draw).
Each of the boxes can have a tile of any type, so there
are 2^n2 pipe dreams of order n.
See <Ref>, left, for an example of a pipe dream of order 6.



A pipe dream (a tiling of the staircase shape) forms
a collection of strands (or pipes) labeled 1 to n from
the row where they start.
A pipe dream is called reduced if
any two pipes cross through each other at most once.


	
	Given a pipe dream D
	that is not necessarily reduced, the reduction of D
	is a unique reduced pipe dream D' obtained as follows:
	starting at the bottom left tile traverse the pipe dream upwards along columns and to the right. For each encountered crossing,
	replace it with a bump if the pipes have already crossed in the already traversed squares.
	The labeling of pipes together with a reduction
	is indicated by colored paths in
	<Ref>, right.



	
	One can associate a permutation w(D)∈ S_n to a pipe dream
	of order n as
	follows. If D is reduced then w(D)^-1_j is the column where
	the pipe j ends up in.[Throughout
	the paper, the column coordinate j increases from left to right,
	and the row coordinate i increases from top to bottom.]
	Equivalently, the column j contains the exiting pipe of color
	w(D)_j. Note that to capture the column number for the pipe j,
	we need the inverse permutation w(D)^-1.

	If D is not reduced, then w(D)
	is the permutation associated to the reduction D' of D.



	Alternatively, the permutation w(D)
	from a non-reduced pipe dream
	can be defined using the Demazure product. We
	refer to <cit.> for details.


Let (n) and (n) be the sets of pipe
dreams and reduced pipe dreams of size n.
For each w∈ S_n,
let (w) and
(w) be, respectively, the sets of pipe dreams and reduced pipe dreams D such that w(D)=w. Note that
#(n)=2^n2, whereas there is no simple
known formula for #(n)
<cit.>. Given
a pipe dream D, let (D) be the set of
coordinates (i,j)
of the cross tiles.
The weight of D is the monomial
(D)∏_(i,j)∈(D) x_i.
For example, for the pipe dream in <Ref>, left, we have
(D)=x_1^3 x_2^2 x_3^2 x_4 x_5.
The following
result is due to
<cit.>, <cit.>, and <cit.>.

	
	For any w∈ S_n, we have

    𝔊_w^β(x_1,…,x_n )  = ∑_D ∈(w)β^#(D)-ℓ(w)(D).

In particular, setting β=0 forbids non-reduced
pipe dreams, so the Schubert polynomial is

    𝔖_w( x) = ∑_D∈(w)(D).





 §.§ Grothendieck random permutations from reduced pipe dreams



Fix p∈[0,1].
Equip the set of all pipe dreams
with a probability measure by independently
placing the tiles in each box:

    -6pt
        < g r a p h i c s >
      with probability p,
    	    -6pt
        < g r a p h i c s >
      with probability 1-p.

In particular, for p=1/2, we have the uniform measure on the set of pipe dreams.

By reducing this random pipe dream as in <Ref>,
we obtain a
random permutation 𝐰∈ S_n
which we call the Grothendieck random permutation (of order n and with parameter p;
we suppress the dependence on n and p in the notation).
The name is justified by a connection with the polynomials 𝔊_w^β=1.
Indeed,
we have for any w∈ S_n:

    (𝐰=w)
    	=
    	∑_D ∈(w) p^(D)(1-p)^elbow(D)
    	=
    	(1-p)^n2𝔊_w^β=1( p/1-p,…, p/1-p),

where the first equality is simply the definition of the measure (<ref>),
and the second immediately follows from <Ref> with β=1.
In <Ref> below, we explain how the
same distribution on permutations arises from
the six-vertex model with domain wall boundary conditions and free-fermion weights (corresponding
to 2-enumeration of Alternating Sign Matrices).

When p=0 or p=1, we almost surely have
𝐰=id or 𝐰=(n,n-1,…,1 ), respectively.
These cases are trivial, and in the rest of the paper we assume that p∈(0,1).
Individual samples of
Grothendieck random permutations with p=4/5 and p=1/2
and a plot constructed from averaging over many samples
are given in
<Ref>.





 §.§ Asymptotics of Grothendieck random permutations



For w∈ S_n,
define its height function by

    H(x,y)#(
    		{ w^-1(x),w^-1(x+1),…,w^-1(n)  }∩{ y,y+1,…,n  }),

where 1≤ x,y≤ n. In terms of the pipe dream as in <Ref>, right,
H(x,y) is the number of pipes of color ≥ x which exit through the positions
≥ y at the top. In the example in <Ref>, right,
we have H(4,3)=2. See also <Ref>
for an interpretation in terms of the permutation matrix of w,
where H(x,y) gives the number of 1s in the rectangle with lower left corner at (x,y).

Let now 𝐰∈ S_n be the Grothendieck random permutation with parameter p∈(0,1)
which we assume fixed.
This makes the height function H(x,y) random, too.
We show that H(x,y) satisfies the law of large numbers,
and characterize its asymptotic fluctuations:


	
	
		
  * There exists a limiting height function 𝗁^∘ such that
			
    lim_n→∞ n^-1 H(⌊ n𝗑⌋,
    				⌊ n𝗒⌋)=𝗁^∘(𝗑,𝗒),
    				    
    				(𝗑,𝗒)∈[0,1]^2,

			where the convergence is in probability.
			The function 𝗁^∘(𝗑,𝗒)
			is
			explicit, see
			(<ref>).
			It is continuous and depends only on p.
			The graph of 𝗁^∘
			is given in
			<Ref>, right.
		
  * The fluctuations of H(x,y) around 𝗁^∘
			belong to the KPZ universality class:
			
    lim_n→∞( H(⌊ n𝗑⌋,
    				⌊ n𝗒⌋)-n𝗁^∘(𝗑,𝗒)/𝗏(𝗑,𝗒) n^1/3≤ r )=F_2(r),     r∈ℝ,

			where F_2
			is the cumulative distribution function of the Tracy–Widom GUE distribution,
			and 𝗏(𝗑,𝗒) is given in (<ref>).
	

<Ref>
justifies
the simulations in
<Ref>. We prove it in
<Ref>
by realizing H(x,y) as an observable of a
discrete time
TASEP with parallel geometrically distributed jumps started from the densely packed
(step) initial configuration.
Here we omit the technical details of this coupling, and refer to
<Ref> in the text.
By employing known techniques
from Integrable Probability and passing back to the
Grothendieck random permutation, we obtain the limit shape and fluctuations.



The law of large numbers in the first part of
<Ref>
means that the Grothendieck random permutations 𝐰∈ S_n
converge to a deterministic permuton.
Recall that permutons are Borel probability measures μ
on [0,1]^2 with uniform marginals, that
is,
μ( [0,1]× [a,b] )=
μ( [a,b]× [0,1] )=b-a.
For details on permutons
and their connection to pattern frequencies in permutations, we refer to
<cit.>,
<cit.>, and the survey
<cit.>.
Our limiting Grothendieck permuton is completely determined by the limiting height function
via

    μ( [𝗒,1]× [𝗑,1] )
    	=𝗁^∘(𝗑,𝗒).

As a corollary of the permuton convergence,
one can obtain law of large numbers for
arbitrary pattern counts in Grothendieck random permutations.
The simplest example is the number of inversions:


	
	Let
	𝐰=𝐰(n)∈ S_n be the Grothendieck random permutations with
	p=1/2.
	We have
	
    lim_n→∞( 𝐰(n) )/ n2=1-π/4.


The limit in (<ref>)
exists for all p∈(0,1) and is expressed through certain integrals
of algebraic functions.
We are able to evaluate it in a closed form only for p=1/2.
We refer to <Ref> and its proof for more
discussion.


	
	In <Ref>, we connect
	Grothendieck random permutations
	to
	2-enumerated Alternating Sign Matrices
	(bijectively corresponding to the six-vertex model with
	domain wall boundary conditions and free-fermion weights, and also to
	domino tilings of the Aztec diamond).
	The connection between Grothendieck polynomials and the
	six-vertex model goes back to
	<cit.>, see also <cit.>.
	Equivalences between 2-enumerated Alternating Sign Matrices,
	six-vertex models, and domino tilings of the Aztec diamond
	are well-known in statistical mechanics and integrable probability,
	see, for example,
	<cit.>, <cit.>, and
	<cit.>, <cit.>, <cit.>.

	Via this connection,
	our asymptotic results (<Ref>)
	can be recast as results about a random permutation coming from the free-fermion
	six-vertex model
	with domain wall boundary conditions.
	Moreover, there is a vast amount of asymptotic results
	about domino tilings and free-fermion six-vertex models,
	and in <Ref>
	we show how some of them lead to further asymptotic properties
	of Grothendieck random permutations.




 §.§ A deformation and random non-reduced pipe dreams



Let us discuss a deformation of the Grothendieck random permutation
by means of a new parameter
q∈[0,1].
Take a pipe dream of order n obtained
by randomly placing the tiles in the same way as in (<ref>)
(depending on p∈ (0,1)). However, instead of reduction (<Ref>),
let us apply a modified (q-randomized) procedure:

	
	Assume that D is a pipe dream
	that is not necessarily reduced.
	For each crossing tile, consider the incoming pipes.
	There are two cases:
	
		
  * If the lower colored pipe enters from the left (and the higher colored one from the bottom),
			then the pipes cross through each other with probability 1.
		
  * Otherwise, if the lower colored pipe enters from the bottom (and the higher colored one from the left),
			then the pipes cross through each other with probability q.
			That is, with probability 1-q we replace the crossing with a bump.
	
	The random q-reduction steps are done independently at each crossing tile
	when
	reading the staircase diagonal by diagonal starting from the bottom left.
	See <Ref> below for a detailed definition
	of the Markov process for q=0, which has the same order of reduction steps.
	Denote the random permutation obtained by the q-reduction of a random pipe dream
	with the measure (<ref>) by 𝐰^(q)∈ S_n.
	See
	<Ref> for simulations.



	
	For q=0, the q-reduction procedure becomes the usual reduction from <Ref>.
	For q>0, the q-reduction no longer tracks whether the pipes
	have crossed before. Instead, it simply applies the local q-reduction rule at each
	crossing tile,
	based on the relative colors of the incoming pipes.


The asymptotic analysis of 𝐰^(q) for q∈ (0,1]
may be performed similarly to the case q=0.
The underlying particle system is a certain
q-deformation of TASEP which allows both left and right jumps (governed by different rules).
Setting q=1 removes the asymmetry, which changes the
normalization in the law of large numbers and fluctuations (as well as
the fluctuation distribution).
This behavior
parallels the different
scales of laws of large numbers
in the
KPZ and the
Edwards–Wilkinson (EW) universality classes.
The KPZ class asymptotics arise in TASEP and its asymmetric deformations,
while symmetric versions of TASEP fall into the EW class.
We refer to
<cit.> for a detailed discussion of these two universality classes.

In this paper, we mainly focus on the case q=0, and obtain
results outlined in <Ref>.  In
<Ref>, we also treat the case q=1.  In
the latter model, the q-reduction does not change the pipe
dream at all. Probabilistically, this is the most natural
way to associate a permutation to a random pipe dream with
distribution (<ref>). On the other
hand, the q=1 model apparently lacks a rich algebraic
underpinning.

Finding the asymptotic number of inversions of
𝐰^(1)=𝐰^(1)(n)
∈ S_n as n→∞ was posed as an open
problem by Colin
Defant at the Richard Stanley's 80th birthday
conference.[Conference “The Many Combinatorial Legacies of Richard P. Stanley: Immense Birthday Glory of the Epic Catalonian Rascal,” <https://www.math.harvard.edu/event/math-conference-honoring-richard-p-stanley/>. Accessed: 07/23/2024.]
We show
the following
asymptotics:

	
	For p∈[0,1) and any ε>0,
	the expected number of inversions satisfies for all sufficiently large n:
	
    2/3√(π) (1-ε)  n^3/2√(p/1-p)≤𝔼[(𝐰^(1)(n))] ≤4/3√(π) (1+ε)  n^3/2√(p/1-p).

	Moreover,
	the random permutations 𝐰^(1)(n) converge in distribution
	to the identity permuton supported
	on the diagonal of [0,1]^2.


Based on simulations (see <Ref> for examples)
and supporting numerics, we make
further conjectures about the behavior of 𝐰^(q).
(<ref>):

	
	For any p∈[0,1), we have convergence in probability:
	
    n^-3/2(𝐰^(1))→ϰ√(p/1-p),     n→∞,

	where ϰ is a constant independent of p whose value is near 0.53.

<Ref> implies that
ϰ (if it exists) must satisfy
2/(3√(π))≤ϰ≤ 4/(3√(π)), and the approximate value 0.53
is conjectured based on numerical data.

We also expect that as p=p(n)→1 with n, the random
permutations 𝐰^(1)(n) converge in
distribution to a nontrivial deterministic permuton (which
depends on the speed and the parameters in p(n)→1).



In the intermediate cases q∈(0,1), one has to make sure that pipes with larger numbers
do not get stuck close to the diagonal boundary. This leads to a
conjectural restriction pq≤1/2.
Let H^(q)(x,y) be the height function of 𝐰^(q),
defined in the same way as in <Ref> above.

	
	For any fixed p,q∈(0,1) with pq≤1/2, the random permutations
	𝐰^(q)(n)∈ S_n converge to a permuton
	determined by a height function
	𝗁^(q)(𝗑,𝗒) on [0,1]^2.
	That is, we have the limit in
	probability:
	
    lim_n→∞ n^-1 H^(q)(⌊ n𝗑⌋,
    		⌊ n𝗒⌋)=𝗁^(q)(𝗑,𝗒),
    		    
    		(𝗑,𝗒)∈[0,1]^2.

	When pq>1/2, the random
	permutations 𝐰^(q)(n) converge to the identity permuton.

When pq≤1/2, fluctuations of H^(q)(x,y) around 𝗁^(q) should
be governed by one of the laws appearing in the
KPZ universality class in the presence of a boundary
(Tracy–Widom GUE distribution,
its GOE/GSE counterparts and crossovers,
or Gaussian fluctuations on scale n^1/2).
We refer to <cit.>, <cit.> for further discussion of
fluctuations in half-space models from the KPZ universality class.






 §.§ Asymptotics and maxima of Grothendieck principal specializations



We are interested in the following principal specializations of the
Grothendieck polynomials:

    Υ_w(β) 𝔊_w^β (1,1,…,1_n),
    	     w∈ S_n.

In particular, Υ_w(0) is the principal specialization of the Schubert polynomial.
For β=1, these quantities are the (unnormalized) probability weights
of the Grothendieck random permutation 𝐰 with p=1/2,
see (<ref>).

Set

    v_n(β)∑_w ∈ S_nΥ_w(β),    
    	u_n(β)max_w∈ S_nΥ_w(β).

From <Ref>, we have:

    v_n(1)=2^n2,      u_n(1)=2^n2-o(n^2),      n→∞.

The first equality is exact, while the second one is asymptotic.
This asymptotic behavior
follows from
the cardinality
n!∼ e^nlog n+O(n)≪ 2^n2 of the set over which we take the maximum.

For u ∈ S_k, w ∈ S_m, we denote

    u × w  (u(1), …, u(k), w(1)+k, …, w(m) + k)∈ S_k+m.

For a composition b = (b_ℓ, …, b_1) of n = b_1 + ⋯ + b_ℓ, the layered permutation w(b) ∈ S_n is defined as follows:

    w(b) = w_0(b_ℓ) ×⋯× w_0(b_1),

where w_0(k) is the full reversal permutation of order k.
Denote by L_n⊂ S_n the subset of layered permutations,
and let

    u_n'(1)max_w∈ L_nΥ_w(1).





In <Ref>, we prove that
on layered permutations, the
β=1 Grothendieck polynomials
attain their asymptotic maximum:


	
There are sequences of layered permutations w(b^(n))∈ S_n so that

    lim_n →∞1/n^2log_2
    Υ_w(b^(n)) (1)
    = 1/2.


<Ref> implies that in leading order,
u_n'(1) asymptotically behaves in the same way as
v_n(1) and u_n(1) (<ref>).

Explicit constructions of such sequences of layered
permutations are given in <Ref>. In particular, we
can take the parts of the compositions b^(n) = (…, b_2, b_1)
to be
geometric b_i ∼
(1- α)α^i-1 n for any α∈ [1/√(2), 1).
See
<Ref> for an illustration.
Note, however that we do not know in the limit what
compositions b of size n yield the global maximum of
Υ_w(b) over all layered permutations. For
Schubert polynomials, the analogous question was settled in
<cit.>. We present numerics
for principal specializations of the β=1 Grothendieck polynomials in
<Ref>, and explain the difference between the (numerically)
optimal layered permutations and the ones constructed in the proof of <Ref>.

Regarding Stanley's problem on permutations achieving the
maximal Schubert specialization u_n(0) (the case
β=0), the Merzon–Smirnov conjecture
<cit.> states that the maximum
is attained on layered permutations. On one hand, our
<Ref>
establishes that an asymptotic analog of this conjecture holds for
the β=1
Grothendieck polynomials.
On the other hand, the typical shape of Grothendieck
random permutations
(<Ref>, center and right; see also
<Ref>)
suggests that the
global maximum u_n(1)
may be achieved on permutations whose shape is far from layered.
In <Ref>, we obtain new bounds on the quantities
v_n(β) and u_n(β) for general values of β,
but do not consider the relation
of maximal principal specializations to layered permutations.



 §.§ Outline


In the Introduction, we defined the model and formulated our main results.
In <Ref>, we explain how the Grothendieck random permutation
𝐰 from <Ref>
is sampled by running a Markov chain, more precisely, the colored stochastic six-vertex model.
In <Ref>, we connect the vertex model
to a discrete time TASEP with parallel update and geometrically distributed jumps.
This process is well-studied within Integrable Probability,
which leads to law of large numbers and Tracy–Widom fluctuation
results.
In <Ref>, we recast the asymptotic results about
TASEP into
limiting properties of
Grothendieck random permutations.
In <Ref>, we consider a variant of our
model coming from non-reduced pipe dreams (described in <Ref>).
In <Ref>, we
address the question of asymptotics of  principal specializations
of β=1 Grothendieck polynomials, and show that they are asymptotically the largest
on layered permutations.
We also obtain new bounds on the principal specializations for general β,
based on an interpretation of Grothendieck polynomials in terms of
Alternating Sign Matrices.
In <Ref>, we discuss further directions and a few
open problems.


In <Ref>, we provide tables of maximal  principal specializations of β=1 Grothendieck polynomials on layered permutations.
In <Ref>, we provide the steepest descent computations
leading to the asymptotics in TASEP. While these computations are standard by
now, we include them to justify the exact values of the constants
in TASEP limit shape and fluctuations.




 §.§ Acknowledgments


We are very grateful to Jehanne Dousse for her valuable insights in the development of this project and doing experiments in Sage. We thank Igor Pak for bringing us together and initiating our collaboration.
We thank Daoji Huang, Mark Shimozono and Tianyi Yu for sharing an early version of <cit.>. We would also like to thank Dave Anderson, Jacopo Borga, Filippo Colomo, Colin Defant, Philippe DiFrancesco, Rick Kenyon, Matthew Nicoletti, Mikhail Tikhonov, and Anna Weigandt for helpful discussions.

The project was begun at the SQuaRE “Young Tableaux Asymptotics”
at the American Institute of Mathematics. The authors thank
AIM for providing a supportive and mathematically rich
environment.
Part of this research was performed in Spring 2024 while
some of the authors were
visiting the
program
“Geometry, Statistical Mechanics, and Integrability”
at the Institute for Pure and Applied Mathematics
(IPAM), which is supported by the NSF grant DMS-1925919.

AM was partially supported by the NSF grant DMS-2154019, the NSERC Discovery Grant
RGPIN-2024-06246, and the FRQNT Team grant 341288. GP was partially supported by the NSF grant CCF-2302174. LP was partially supported by the NSF grant DMS-2153869 and the Simons Collaboration Grant for Mathematicians 709055. DY was partially supported by the  grant.



§ FROM PIPE DREAMS TO THE COLORED STOCHASTIC SIX-VERTEX MODEL





 §.§ Colored stochastic six-vertex model



Introduce the vertex weights

    w_p(a,b;c,d)
    =
    w_p
    (
    	[baseline=-3,scale=.7,very thick]
    			[fill] (0,0) circle [radius=0.025];
    			 (0.5,0) – (0.05,0);
    			 (-0.5,0) – (-0.05,0);
    			 (0,0.05) – (0, 0.5);
    			 (0,-0.05) – (0,-0.5);
    			 at (.25,-.4) a;
    			 at (-.7,0) b;
    			 at (-.25,.4) c;
    			 at (.7,0) d;
    ),
 where
a,b,c,d∈{0, 1,…,n  }.
Here 0 represents
the absence of a pipe, and
positive numbers indicate the
pipes' colors.
We view (a,b) and (c,d) as incoming and outgoing pipes, respectively.
The weights are
defined as follows:

    w_p(a,a;a,a)    =1;
    		
    
    		w_p(b,a;b,a)   =p,       w_p(b,a;a,b)=1-p;
    		
    
    		w_p(a,b;a,b)   =0,       w_p(a,b;b,a)=1,

where 0≤ a<b≤ n.
The weights of all other configurations not listed in (<ref>)
are zero. Let us indicate a few crucial properties of the weights:

	
  * The weights conserve the pipes:
		w_p(a,b;c,d)=0 unless {a,b}={c,d} as sets.
	
  * The weights are stochastic:
		
    w_p(a,b;c,d)≥0,    ∑_c,d=0^n w_p(a,b;c,d)=1   for all a,b.




	The weights w_p come from the
	fundamental stochastic R-matrix
	for the quantum group U_q(𝔰𝔩_n+1)
	<cit.>, see also <cit.>
	for a brief overview.
	For the application to random pipe dreams, we specialize
	the quantum parameter q to zero
	(equivalently, to infinity in the normalization of <cit.> — this choice
	depends only on
	the order in which colors correspond to basis vectors of
	the fundamental representation of 𝔰𝔩_n+1).
	The remaining parameter p is related to the spectral parameter
	in an integrable vertex model.




 §.§ Matching random pipe dreams to the stochastic vertex model



Attach coordinates
(i,j)∈ℤ_≥1 ^2
to boxes of the
staircase shape, with i increasing down,
and j increasing to the right.
The staircase shape is then
_n{ (i,j) i+j≤ n }.

Place a stochastic vertex with the weight w_p at each box (i,j)∈_n.
Let the initial condition along the left boundary of _n be
the rainbow one, with colors 1 to n from top to bottom.
Then we can sample a random configuration
of pipes as in <Ref>, right,
by running a discrete time Markov chain with time τ=j-i, where
-(n-1)≤τ≤ n-1. At each step τ→τ+1,
the configuration with j-i≤τ
is already sampled, which determines the incoming colors at all boxes
(i,j)∈_n with j-i=τ+1. The next
step τ→τ+1 consists in an independent update of the outgoing colors
at all boxes (i,j)∈_n with j-i=τ+1,
using the stochastic vertex weights
w_p(a,b;·,·)
(<ref>), 1≤ a,b≤ n.
Here we view each w_p(a,b;·,·) as a probability distribution on possible outputs,
(a,b) or (b,a).
Reading off the outgoing colors at the top boundary of _n,
we arrive at a random permutation 𝐰∈ S_n.


	
	The random permutation 𝐰∈ S_n obtained from the colored stochastic six-vertex model
	as described above has the same distribution as the Grothendieck random permutation
	defined in <Ref>.


	The Markov evolution of the stochastic vertex model
	is equivalent to a simultaneous
	exploration of a random pipe dream (that is, determining the state
	-2pt
    < g r a p h i c s >

	or
	-2pt
    < g r a p h i c s >

	at each box),
	and its reduction by following the pipes.
	Indeed, in
	the evolution of the stochastic vertex model,
	in each box (i,j), two strands of different colors meet as incoming pipes.
	There are two possibilities:
	
		
  * If the pipe of the lower numbered color
			is below, then these pipes have already met and crossed through each other
			(which is allowed only once in a reduced pipe dream).
			Thus, regardless of the state
			-2pt
    < g r a p h i c s >

			or
			-2pt
    < g r a p h i c s >

			at (i,j), the pipes must bump off each other.
			In
			(<ref>),
			this corresponds to the vertex weight w_p(a,b;b,a)=1, where 1≤ a<b≤ n.
		
  * If the pipe of the lower numbered color is to the left, then the pipes have not yet crossed
			through each other (but they may have bumped off at an elbow).
			Then, we place the tile
			-2pt
    < g r a p h i c s >

			or
			-2pt
    < g r a p h i c s >

			at (i,j) with probability p or 1-p, respectively.
			After placing the tile,
			the strands deterministically follow the paths drawn on it.
			The choice of the tile is equivalent
			to using the stochastic vertex weights
			w_p(b,a;b,a)=p or w_p(b,a;a,b)=1-p, where 1≤ a<b≤ n.
	
	We see by induction that the vertex model
	(<ref>)
	produces the same random permutation
	as reducing the random pipe dream.




 §.§ Height function and color forgetting



Let x,y∈{ 1,…,n  }.
For a fixed pipe dream D and the corresponding permutation w=w(D)
(see <Ref>),
define the permutation height function as

    H(x,y)
    		   #{pipes of colors ≥ x
    		which exit through positions j≥ y at the top}
    =   #(
    			{ w^-1(x),w^-1(x+1),…,w^-1(n)  }∩{ y,y+1,…,n  }).

Observe that H(x,y) depends only on the permutation
w and not on the structure of crossings and bumps in the pipe dream.
For a fixed w, the function H(x,y) decreases in x and y.
When the pipe dream and the permutation 𝐰
are random, the height function H(x,y) becomes a random variable.

In the permutation matrix of w∈ S_n,
H(x,y) is the number of entries in the rectangle
[y,n]×[x,n]. Recall that the pipe of color i exits at w^-1(i), so the
rectangle is transposed.
See <Ref> for an illustration.
This interpretation of H(x,y) implies that

    H(x,y)≤min(n-x,n-y).


In colored stochastic six-vertex models, quantities like H(x,y) are
referred to as colored height functions, cf. <cit.>.




	
	The color-position symmetry <cit.>
	of the colored stochastic six-vertex model
	implies that the distribution of H(x,y) corresponding to the
	Grothendieck random permutation 𝐰
	is symmetric in x,y.
	Indeed, by the color-position symmetry, 𝐰 and 𝐰^-1
	have the same distribution. Therefore,
	
    H(x,y)
    		=
    		∑_i=x^n∑_j=y^n1_𝐰^-1(i)=jd=∑_i=x^n∑_j=y^n1_𝐰(i)=j
    		= H(y,x).

	However, we can access the distribution of H(x,y) directly
	using color forgetting (<Ref> below), and do not
	rely on
	color-position symmetry.


Consider an uncolored (color-blind) stochastic
vertex model
with all pipes of the same color.
We indicate pipes and empty edges by 0 and 1, respectively.
The weights of the color-blind model are defined as

    w_p^∙(0,0;0,0)   =1,     w_p^∙(1,1;1,1)=1,
    		
    
    		w_p^∙(1,0;1,0)   =p,    
    		w_p^∙(1,0;0,1)=1-p,
    		
    
    		w_p^∙(0,1;0,1)   =0,    
    		w_p^∙(0,1;1,0)=1.



	
	Fix 1≤ x≤ n. In the colored stochastic vertex model of
	<Ref>,
	erase all pipes of colors <x,
	and identify all the remaining pipes for colors ≥ x.
	The resulting random configuration of
	uncolored pipes evolves
	according to
	the color-blind stochastic
	vertex model w_p^∙ (<ref>).



	This is a standard property of the colored stochastic six-vertex model,
	see <cit.> and references therein.
	In interacting particle systems,
	this property is known as the basic coupling, cf.
	<cit.>, <cit.>.
	Let us provide an idea of the proof.

	The crucial property is
	that the weights w_p (<ref>)
	depend only on the relative order of colors of the incoming pipes, and not the actual colors.
	Therefore, when we identify all colors ≥ x, the resolution of the pipe
	crossings does not affect the behavior of the color-blind system.
	In detail, one can consider four cases
	for the incoming pipes (a,b),
	1≤ a,b≤ n,
	depending on whether a≥ x and/or b≥ x.
	The resulting probability distribution on possible outputs,
	(a,b) or (b,a), depends only on these cases, and not on
	the actual values of a,b.
	These four cases correspond to the color-blind incoming pipes
	in w_p^∙ (<ref>).


By <Ref>,
the permutation height function H(x,y)
can be identified with an observable of the model
w_p^∙ with initial occupied configuration
{ x,x+1,…,n   } along the left boundary
(that is, the sites {1,…,x-1} are empty).
Indeed, in this color-blind model, H(x,y) is simply
the
number of pipes exiting through the top boundary at positions ≥ y.
We see that under the color forgetting, the color parameter x in H(x,y) became a parameter of the initial configuration.


	One can similarly forget the colors >x, and the resulting
	color-blind system evolves according to different
	weights (obtained from w_p^∙ by swapping 0↔ 1).
	It is more convenient for us to work with the model w_p^∙,
	which we directly relate
	to an interacting particle system in the next <Ref>.




§ FROM VERTEX MODELS TO TASEP





 §.§ TASEP with moving exit boundary




	
	Let k≥ 1.
	Let
	ξ(t)(ξ_1(t)>…>ξ_k(t) )⊂ℤ
	be the
	k-particle
	discrete time TASEP (Totally Asymmetric Simple Exclusion Process)
	having
	parallel updates
	and geometrically distributed jumps.
	In detail, ξ(t), t∈ℤ_≥0, is a discrete time Markov chain
	on particle configurations in ℤ which at each time step t→ t+1 evolves as follows:
	
    ξ_i(t+1)=ξ_i(t)+min( G_i(t+1), ξ_i-1(t)-ξ_i(t)-1 ),    
    		1≤ i≤ k,

	where G_i(t+1) are independent geometric random variables with parameter p∈(0,1),
	that is,
	
    ( G_i(t+1)=m )=(1-p) p^m,     m∈ℤ_≥0.

	The update (<ref>) occurs in parallel for all
	particles 1≤ i≤ k, that is,
	the new positions ξ_i(t+1) depend only on the configuration
	ξ(t) at the previous time step, and new independent random variables.
	By agreement, we have ξ_0(t)≡ +∞,
	so that the first particle ξ_1(t) performs an independent random walk
	with geometrically distributed jumps.
	See <Ref> for an illustration.




Start the TASEP from
the densely packed (also called step)
initial configuration
{ 1,2,…,k  }, that is,
ξ_i(0)=k+1-i, 1≤ i≤ k.
Fix n≥ k, and introduce a moving exit boundary
which starts at n+1/2, and deterministically
moves to the left with speed 1.
The exit time of a particle ξ_i(t) is defined by

    T_exit(i)min{ tξ_i(t)≥ n+1-t }.

For example, if n=k, then the first particle starting at
n exits at time t=1 (but have not exited at the initial
time t=0). The exit times are almost surely ordered:

    1≤ T_exit(1)<T_exit(2)<…<T_exit(k)≤ n.

Note that tor any n≥ k,
all particles have exited the system by the time t=n.
See <Ref> for an illustration and simulations.







Integrability of TASEP with parallel updates, geometric jumps,
and densely packed initial configuration
can be traced back to
<cit.>, see also
<cit.>,
<cit.>,
and <cit.>.
We recall the necessary
integrability
results in <Ref> below
after connecting TASEP to vertex models and
Grothendieck random permutations
in the next <Ref>.



 §.§ Identification with vertex models



Fix n and 1≤ x,y≤ n. Recall the
observables H(x,y)
defined by (<ref>).
That is, H(x,y) are random variables which are functions of the Grothendieck
random permutation (equivalently, of the colored stochastic six-vertex model).
In <Ref>,
we identified H(x,y) with observables of a color-blind vertex model with weights
w_p^∙ (<ref>).


	
	For any 1≤ x,y≤ n and 0≤ h≤ n-x+1, we have
	
    _𝐰(H(x,y)≤ h)
    			   =
    			_TASEP(T_exit(n-x+1-h)≤ y-1)
    			
       =
    			_TASEP(
    				ξ_n-x+1-h(y-1)≥ n-y+2
    			).

	Here
	_𝐰
	corresponds to the Grothendieck random permutation of order n,
	and
	_TASEP
	is the probability distribution of the TASEP
	with k=n-x+1 particles and moving exit boundary.
	In (<ref>)
	we have, by agreement,
	T_exit(m)= 0 and
	ξ_m(t)= +∞ for all m≤ 0.


	The second identity in (<ref>)
	immediately follows from the definition of the exit times
	(<ref>). Let us focus on the first identity.

	By <Ref>, let us consider the color-blind vertex model
	with k=n-x+1 identical pipes entering the left boundary
	of the staircase shape _n
	at locations { x,x+1,…,n  }.
	The event { H(x,y)≤ h } means that
	at most h of these pipes exit the top boundary at positions ≥ y.
	Note that the color-blind model has indistinguishable pipes, and thus the “resolution”
	of their crossings
	does not change the behavior of the system (see the proof of <Ref> for more detail).

	View the horizontal coordinate j
	as time t=0,1,…,n,
	and record the pipes' coordinates as
	
    η_1(t)<…<η_k(t),     t=0,1,…,n .

	The initial condition is η_k-m+1(0)=n-m+1, 1≤ m≤ k.

	At each time step, according to the w_p^∙ (<ref>),
	each pipe first deterministically turns up.
 	When a pipe faces up, it can travel up a further
	distance distributed as a geometric random variable with parameter p,
	see (<ref>).
	However, during a time step t→ t+1, each pipe η_m cannot
	travel further than the previous pipe' location η_m-1(t)
	(this is the exclusion mechanism).
	When a pipe stops moving up, it turns right and waits till the next time step.
	Once a pipe reaches the top boundary, it exits the system.
	See <Ref>, left, for an illustration.



	

	Setting ξ_m(t) n+1-t-η_m(t),
	one readily verifies that the evolution of ξ_1(t)> … > ξ_k(t)
	is the same as that of the TASEP
	with k=n-x+1 particles
	defined in
	<Ref>.
	In particular, subtracting t from n+1-η_m(t)
	eliminates the deterministic up movement of the pipes by one at each time step.

	The top boundary for pipes (as in <Ref>, left)
	becomes the moving exit boundary for TASEP. The event
	{H(x,y)≤ h} for pipes is equivalent to the event
	that at least k-h=n-x+1-h of the TASEP particles
	have exited before time y-1. Because the
	exit times are ordered as in (<ref>),
	we get the desired first identity in (<ref>).




 §.§ Integrability of TASEP



Fix m≥ 0, and let λ=(λ_1≥…≥λ_m≥ 0 )
be a partition with at most m parts.
The Schur symmetric polynomial in m variables indexed by λ is defined as

    s_λ(a_1,…,a_m)[a_i^λ_j+m-j]_i,j=1^m/[a_i^m-j]_i,j=1^m,

where a_i are variables. The denominator is the Vandermonde determinant
∏_1≤ i<j≤ m(a_i-a_j).

Let m,t≥1.
The Schur measure <cit.>
is a probability measure on partitions with at most min(m,t) parts
depending on parameters a_1,…,a_m,b_1,…,b_t. Its probability weights
are defined as

    (λ)∏_i=1^m∏_j=1^t
    	(1-a_ib_j)
    	·
    	s_λ(a_1,…,a_m)
    	s_λ(b_1,…,b_t).

When m=t=1, the weights (<ref>) define the geometric distribution
on λ=(λ_1≥0) with parameter a_1b_1.
In general, the form of the normalizing constant in (<ref>)
follows from the Cauchy summation identity for Schur polynomials.
The infinite sum
for the normalizing constant runs
over all λ of length ≤min(m,t) and converges if |a_ib_j|<1 for all i,j.

The next statement connects TASEP to Schur measure,
which provides the necessary integrability structure for the former.


	
	For any t,m≥1, the displacement of the m-th particle
	in the TASEP with parallel updates, geometric jumps with parameter p,
	and densely packed initial configuration (as in <Ref>)
	has the same distribution as λ_m, the last part of a partition
	under the Schur measure
	(<ref>)
	with a_1=…=a_m=1 and b_1=…=b_t=p:
	
    ξ_m(t)-ξ_m(0)d=λ_m,
    		    λ∼
    		(1-p)^m t s_λ(1,…,1 _m)
    		s_λ(p,…,p _t).



<Ref> is a well-known
result in Integrable Probability.
It follows either from the
Robinson–Schensted–Knuth (RSK) correspondence with column insertion,
or by representing the TASEP as a
marginally Markovian evolution of a process on the space of semistandard Young tableaux
(equivalently, interlacing arrays / Gelfand–Tsetlin patterns).

For the simpler continuous time TASEP (when each particle waits an exponentially
distributed time and jumps to the right by 1 if the destination
is unoccupied), the connection to Schur measures is
exploited in the celebrated work <cit.>
via last-passage percolation, which is related to the row RSK.
The RSK approach can be traced back to <cit.>.
Dynamics on interlacing arrays are constructed later in
<cit.> using a non-RSK approach.

For our discrete time TASEP with geometric jumps,
the RSK was explicitly used in
<cit.>
to establish <Ref>.
A systematic treatment of discrete time
dynamics on interlacing arrays connected to Schur measures
leading to various TASEPs and PushTASEPs may be found in
<cit.>.
It includes
RSK-type and other dynamics on interlacing arrays under one roof,
as well as a generalization of the Schur structure to the level of q-Whittaker
symmetric polynomials.



 §.§ Determinantal structure



The connection
to Schur measures (<Ref>)
allows to extract law of large numbers and
fluctuation results for the TASEP particles.
The key property of Schur measures which makes this possible is
the fact that they form a determinantal point process.

Fix m,t≥ 1, and define a random point configuration
X(λ){λ_j+m-j}_j=1^m on ℤ_≥0,
where λ is distributed as (<ref>).
If λ has less parts than m, append it by zeros.
These zeros translate into a part of X(λ) of the form
{0,1,…,l } for some l.


	
	The random point configuration X(λ) on ℤ_≥0 is a
	determinantal point process. This means that for
	any r≥1 and pairwise distinct points u_1,…,u_r ∈ℤ_≥0,
	the correlation functions have the form
	
    (X(λ) contains all of the points u_1,…,u_r )=
    		[ K(u_i,u_j) ]_i,j=1^r.

	The correlation kernel K is given by a double contour integral:
	
    K(u_1,u_2)=1/(2π𝐢)^2dz  dw/z-ww^u_2-m/z^u_1-m+1(1-p/z)^m/(1-z)^t(1-w)^t/(1-p/w)^m,
    		    
    		u_1,u_2∈ℤ_≥0,

	where the contours are positively oriented simple closed curves satisfying
	p<|w|<|z|<1.


<Ref> is due to
<cit.>,
see also
<cit.>.
General discussions and many examples of determinantal point processes
may be found
in <cit.>, <cit.>,
or
<cit.>.




 §.§ Asymptotics of TASEP



Determinantal structure powers the asymptotic behavior of
the TASEP particles.
We are interested in
probabilities of the form

    _TASEP(ξ̅_m(t)≥ u),
    	    where  ξ̅_m(t)ξ_m(t)-ξ_m(0).

Indeed, since
<Ref>
reduces the distribution of H(x,y) to
probabilities of the form (<ref>),
we can forget about the moving exit boundary in TASEP.
Indeed, the parameter n of the moving boundary is absorbed into m and u.



Let the parameters
t,m,u grow to infinity proportionally to each other:

    m=⌊ L𝗆⌋ ,     
    	t=⌊ L𝗍⌋ ,     
    	u=⌊ L𝗎⌋ ,
    	      L→∞.

First, let us heuristically discuss the law of large numbers and fluctuations
of
_TASEP(ξ̅_m(t)≥ u),
without providing exact constants.
The latter are given in <Ref> below,
and we formulate the asymptotic results in <Ref>.
The proof of <Ref> using the determinantal structure
is a standard application of the steepest descent method for double contour integrals.
We provide the computations (leading to the constants in <Ref>)
in
<Ref>.



At the “hydrodynamic” scale (<ref>),
TASEP has a limit shape
(first observed in the continuous time TASEP in <cit.>, see also
<cit.>). That is, there exists
a function 𝖼(𝗆,𝗍) such that
for any ε>0, we have

    lim_L→∞_TASEP(ξ̅_m(t)≥ L𝖼(𝗆,𝗍)+ε)
    	=0,
    	    lim_L→∞_TASEP(ξ̅_m(t)≥ L𝖼(𝗆,𝗍)-ε)
    	=1.

In other words, we have convergence in probability:

    lim_L→∞
    	L^-1ξ̅_⌊ L𝗆⌋(⌊ L𝗍⌋ )
    	=𝖼(𝗆,𝗍).

Let us note a certain a priori property of the limit shape:

	
	If 𝗍<𝗆/p, we have 𝖼(𝗆,𝗍)=0.


	This follows from the fact that the leftmost hole in the TASEP configuration
	(which starts immediately to the right of ξ_1(0) at time t=0)
	travels left by at most 1 at each time step.
	The probability that the hole moves is p=(G>0),
	where G is the geometric random variable (<ref>).
	Therefore, if 𝗍<𝗆/p, then the hole has not reached the
	m-th particle by time t with probability 1-e^-C L for some C>0.
	This implies the claim.


Next, since TASEP belongs to the Kardar–Parisi–Zhang (KPZ) universality class
<cit.>, we expect
fluctuations of order L^1/3 provided that 𝗍>𝗆/p.
More precisely,
modifying m and u on the scale
L^1/3 probes the scaling window in the law of large numbers
(<ref>). That is,
there exist constants
𝗏_1(𝗆,𝗍),
𝗏_2(𝗆,𝗍)>0, such that

    lim_L→∞_TASEP( ξ̅_⌊ L𝗆- L^1/3α𝗏_1(𝗆,𝗍)⌋
    		(⌊ L𝗍⌋ )≥
    	L𝖼(𝗆,𝗍)-
    	L^1/3β𝗏_2(𝗆,𝗍))
    	=F_2( α +β)

for all α,β∈ℝ.
Here, F_2 denotes the cumulative distribution function of
the Tracy–Widom GUE distribution. The plus signs
by α and β in the right-hand side of (<ref>)
are straightforward from the monotonicity of the pre-limit probabilities.

The distribution function F_2
was first discovered
in connection with the fluctuations of the largest
eigenvalue in large complex Hermitian random matrices
exhibiting unitary symmetry <cit.>.
Subsequently,
F_2 was put into a larger class of limiting distributions arising in random
growth models and interacting particle systems, known as the KPZ universality class.
We refer to the surveys <cit.>,
<cit.>,
<cit.> for a detailed exposition.
The appearance of F_2 in the fluctuations (out of several
candidates in the KPZ universality class)
is a feature of the step initial
configuration in our TASEP.


	
	Let 𝗆,𝗍>0. Define
	
    𝖼(𝗆,𝗍)
    		
    			0,
    			   if 𝗍≤𝗆/p;
    			
    (√(p𝗍)-√(𝗆))^2/1-p,
    			   if 𝗍≥𝗆/p.

	If 𝗍>𝗆/p, set
	
    𝗏_1(𝗆,𝗍)√(p)𝗆^1/3/𝗍^1/6(√(𝗍/p)-√(𝗆))^2/3/(√(p𝗍)-√(𝗆))^1/3,
    		    𝗏_2(𝗆,𝗍)√(p)
    		(√(p𝗍)-√(𝗆))^2/3
    		(√(𝗍/p)-√(𝗆))^2/3/(𝗆𝗍)^1/6(1-p)
    		.




	
	Let the constants 𝖼(𝗆,𝗍) and 𝗏_1,2(𝗆,𝗍) be
	given in <Ref>. Then
	
		
  * The law of large numbers (<ref>) holds for all
			𝗆,𝗍>0.
		
  * The Tracy–Widom fluctuation result (<ref>) holds
			under the condition 𝗍>𝗆/p>0.
	

The proof of <Ref> is given in <Ref>.



§ LIMIT BEHAVIOR OF GROTHENDIECK RANDOM PERMUTATIONS





 §.§ Law of large numbers for the permutation height function



In this section, we obtain the asymptotic behavior
of Grothendieck random permutations of growing order
via the
the pre-limit identity in distribution (<Ref>)
and
TASEP asymptotics (<Ref>).
As a result, we complete the proof of <Ref>
formulated in the Introduction.
First, we focus on
computations involving the law of large numbers, and
then add the fluctuation terms.
By <Ref>, we have

    _𝐰(H(x,y)≤ h)
    	=
    	_TASEP(
    		ξ_n-x+1-h(y-1)≥ n-y+2
    	),

where H is the height function of the Grothendieck random
permutation 𝐰, and
ξ_i(t) are the TASEP particles
with
the step initial configuration
ξ_i(0)=n-x-i+2, 1≤ i≤ n-x+1.
Therefore,

    _𝐰(H(x,y)≤ h)
    	=
    	_TASEP(
    		ξ̅_n-x+1-h(y-1)≥ n-y+1-h
    	),

where ξ̅_i(t)=ξ_i(t)-ξ_i(0) is the displacement.

Let h=⌊ n𝗁⌋, where 𝗁 is the
scaled permutation height function.
Let also
x=⌊ n𝗑⌋, y=⌊ n𝗒⌋,
where 𝗑,𝗒∈[0,1] are the scaled coordinates in the permutation matrix.
Note that
by (<ref>), we have
0≤𝗁≤min(1-𝗑,1-𝗒).

For each 𝗑,𝗒,𝗁, the
asymptotic location
of n^-1ξ̅_n-x+1-h(y-1) is determined
from the TASEP law of large numbers (<ref>), (<ref>).
Namely, this location is equal to

    𝖼(1-𝗑-𝗁,𝗒)=
    	(√(p𝗒)-√(1-𝗑-𝗁))^2/1-p1_p𝗒≥ 1-𝗑-𝗁
    	.

There are two cases.
If

    𝖼(1-𝗑-𝗁,𝗒)> 1-𝗒-𝗁,

then the probability (<ref>) goes to 1.
If the inequality is reversed, then this probability goes to 0.
As 𝗁 increases,
the left- and the right-hand sides of (<ref>)
increase and decrease, respectively.
Therefore, we should define

    𝗁^∘(𝗑,𝗒)inf{𝗁∈[0,min(1-𝗑,1-𝗒)]𝖼(1-𝗑-𝗁,𝗒)> 1-𝗒-𝗁}.

If the set in (<ref>) is empty, we
take the maximal possible value,
𝗁^∘(𝗑,𝗒)=min(1-𝗑,1-𝗒).

To describe 𝗁^∘ more explicitly,
introduce the northeast part of the ellipse

    ℰ_p{
    		(𝗑,𝗒)
    		
    		(𝗒-𝗑)^2/p+(𝗒+𝗑-1)^2/(1-p)=1,		1-p≤𝗑≤ 1
    	}⊂ [0,1]^2.





	
	The function
	𝗁^∘
	(<ref>) is given by
	
    𝗁^∘(𝗑,𝗒)
    		=
    		
    			1-𝗒,   (𝗑,𝗒)∈𝒜{ 0<𝗑<1-p, 𝗑/(1-p)<𝗒<1 }
    			;
    
    			1-𝗑,   
    			(𝗑,𝗒)∈ℬ{ 0<𝗑<1, 0<𝗒<(1-p)𝗑}
    			;
    
    			1+
    			2/p√((1-p) 𝗑𝗒)
    			-𝗑+𝗒/p
    			,   (𝗑,𝗒)∈𝒞{.380<𝗑<1, (1-p)𝗑<𝗒<𝗑/(1-p),
    			
    (𝗑,𝗒)belowℰ_p}
    			;
    
    			0,   
    			(𝗑,𝗒)∈𝒟{(𝗑,𝗒) above ℰ_p}
    			.

	It
	is continuous on [0,1]^2.

See <Ref>, left, for an illustration
of the subsets 𝒜,ℬ,𝒞,𝒟
in (<ref>). The graph of the function
𝗁^∘ is given in
<Ref>, right.

	In 𝒜 and 𝒞,
	we have
	𝖼(1-𝗑-𝗁^∘,𝗒)= 1-𝗒-𝗁^∘.
	Solving this for 𝗁^∘ leads to the result.
	In 𝒟, we have
	𝖼(1-𝗑-𝗁,𝗒)> 1-𝗒-𝗁
	for all 𝗁≥0, so 𝗁^∘(𝗑,𝗒)=0.
	In ℬ,
	we have
	𝖼(1-𝗑-𝗁,𝗒)< 1-𝗒-𝗁
	for all 0≤𝗁≤ 1-𝗑.
	This means that
	the set over which we take the infimum in
	(<ref>) is empty.
	This implies that
	𝗁^∘(𝗑,𝗒)=
	1-𝗑 in ℬ, which
	completes the proof.


<Ref>
and identity (<ref>)
immediately imply the law of large numbers for the
height function:

    lim_n→∞n^-1
    	H
    	( ⌊ n𝗑⌋,
    	⌊ n𝗒⌋)
    	=𝗁^∘(𝗑,𝗒),
    	    𝗑,𝗒∈[0,1],

with convergence in probability.
This completes the proof of the first part of
<Ref>
from the Introduction.



 §.§ Properties of the limiting permuton



The law of large numbers
(<ref>)
implies that the Grothendieck random permutations 𝐰∈ S_n
converge, as n→∞, to a (deterministic) permuton
prescribed by
𝗁^∘(𝗑,𝗒).
Recall that a permuton is a probability measure
on [0,1]^2 with uniform marginals.
We refer to
<cit.>,
<cit.>,
<cit.> for detailed treatment of permutons.

In detail, the permuton is connected to the height function as

    𝗁^∘(𝗑,𝗒) =
    	( X>𝗑, Y>𝗒),
    	    𝗑,𝗒∈[0,1],

where (X,Y)∈[0,1]^2 is a random point distributed according to the permuton.
Note that both X and Y are uniformly distributed on [0,1],
while their joint distribution is nontrivial.
Indeed, from (<ref>)
if follows that
𝗁^∘(𝗑,0) = 1-𝗑,
𝗁^∘(0,𝗒) = 1-𝗒,
as it should be for uniform marginals of a permuton.

Let is find the part of the permuton
that is concentrated on
the curve
ℰ_p (<ref>)
(this concentration is visible in simulations, see <Ref>).
For small ε>0, we have

    𝗁^∘(𝗑,𝗒-ε)-𝗁^∘(𝗑,𝗒)
    	=( X>𝗑, 𝗒-ε<Y≤𝗒).

Dividing this by ε=(𝗒-ε<Y≤𝗒),
taking the limit as ε→0,
and passing to the complement event,
we obtain the conditional cumulative distribution function (cdf) of X given Y=𝗒:

    (X≤𝗑| Y = 𝗒)=
    	1+
    	∂/∂𝗒𝗁^∘(𝗑,𝗒)=
    	
    		0,    (𝗑,𝗒)∈𝒜;
    
    		1,    (𝗑,𝗒)∈ℬ∪𝒟;
    p-1p+√((1-p)𝗑𝗒)p𝗒,    (𝗑,𝗒)∈𝒞.

This function has a jump discontinuity along the ellipse ℰ_p (<ref>),
that is, at the point

    𝗑_p(𝗒)
    	
    	p+𝗒
    	-2 p 𝗒+
    	2 √((1-p)  (1-𝗒) p𝗒),
    	    𝗒>1-p.

We conclude:

	
	Let 𝗒>1-p.
	The unique discontinuity of the conditional distribution of X given Y=𝗒 is
	an atom
	at 𝗑=𝗑_p(𝗒), and the atom's mass is
	
    ( X = 𝗑_p(𝗒)| Y = 𝗒)=
    			1/p
    			-[
    				(1-p) (p+𝗒
    			-2 p 𝗒+
    			2 √((1-p)  (1-𝗒) p𝗒))
    	]^1/2/p√(𝗒),

	where
	1-p<𝗒≤ 1.


In particular, from (<ref>) we have
( X = 1-p| Y = 1)
=1, as it should be.


	
	While conditionally on Y=𝗒 (where 𝗒>1-p),
	the distribution of X has an atom,
	the permuton (that is, the joint distribution of X and Y)
	does not have atoms. This follows from the fact that
	𝗁^∘(𝗑,𝗒)
	(<ref>) is continuous on [0,1]^2.


As an application, let us consider the law of large numbers of the number of inversions
(𝐰)#{i<j𝐰(i)>𝐰(j)}, where
𝐰∈ S_n is the Grothendieck random permutation.
By a result of
<cit.> (see also Theorem 1 in the survey <cit.> or
<cit.>),
the convergence of 𝐰∈ S_n to a permuton
is equivalent to the convergence (in probability) of all pattern counting statistics
of 𝐰.
The limiting pattern counts are determined by the permuton.
In particular,
for inversions, we have

    lim_n→∞(𝐰)/n2=
    	γ_p
    	[ (X_1>X_2,  Y_1<Y_2)  or  (X_1<X_2,  Y_1>Y_2) ],
    	     p∈(0,1),

where (X_1,Y_1) and (X_2,Y_2) are independent
points of [0,1]^2
sampled from the limiting permuton.


	
	We have γ_1/2=1-π/4.

The constant γ_p for general p can be expressed
as a sum of a double and a single
integral of algebraic functions,
and we were not able to evaluate it in closed form except for p=1/2.
The integrals for general p are given in the proof below.

	Fix (𝗑,𝗒)∈[0,1]^2, and let (X,Y)∈[0,1]^2
	be a random point sampled from the permuton.
	We have
	
    [ (X>𝗑,  Y<𝗒)  or  (X<𝗑,  Y>𝗒) ]
    		=1-𝗁^∘(𝗑,𝗒)-
    		𝗁^∙(𝗑,𝗒),

	where 𝗁^∘(𝗑,𝗒)=(X>𝗑,  Y>𝗒) is given by
	(<ref>), and
	
    𝗁^∙(𝗑,𝗒)(X≤𝗑,  Y≤𝗒).

	In (<ref>),
	we used the absence of atoms
	(<Ref>) and
	ignored the difference between strict and weak inequalities.
	The function 𝗁^∙ can be computed using the
	conditional distribution (<ref>):
	
    𝗁^∙(𝗑,𝗒)=
    		∫_0^𝗒(X≤𝗑| Y = w) dw=
    		𝗑,    (𝗑,𝗒)∈𝒜;
    𝗒,    (𝗑,𝗒)∈ℬ;
    p-1/p(𝗑+𝗒)+
    			2/p√((1-p)𝗑𝗒),
    			    (𝗑,𝗒)∈𝒞;
    𝗑+𝗒-1,    (𝗑,𝗒)∈𝒟.


	By (<ref>),
	the constant γ_p (for general p) is given by
	
    γ_p=
    		𝔼[ 1 - 𝗁^∘(X,Y) - 𝗁^∙(X,Y) ],

	where the expectation is taken with respect to the permuton.
	To evaluate this expectation, we use the conditional density
	of X given Y=y which is obtained from
	(<ref>):
	
    f_X| Y=𝗒(𝗑)=
    		1/2p√(1-p/𝗑𝗒)1_(𝗑,𝗒)∈𝒞.

	Recalling that for Y=𝗒>1-p the conditional distribution of X has an atom,
	we can express the expectation as follows:
	
    γ_p=
    			1   -
    			∬_𝒞(
    				𝗁^∘(𝗑,𝗒)+𝗁^∙(𝗑,𝗒)
    			)
    			f_X| Y=𝗒(𝗑) d𝗑 d𝗒
       -
    			∫_1-p^1
    			( 𝗁^∘(𝗑_p(𝗒),𝗒)+
    				𝗁^∙(𝗑_p(𝗒),𝗒)_𝗑_p(𝗒)+
    				𝗒-1)
    			( X=𝗑_p(𝗒)| Y=𝗒) d𝗒.

	For p=1/2, a
	computer algebra system
	like 
	is able to find explicit anti-derivatives, and
	evaluate these integrals.
	We have:
	
    ∬_𝒞4  √(2𝗑𝗒) - 3 𝗑 - 3 𝗒 + 1/√(2𝗑𝗒)  d𝗑 d𝗒=π+2/8,

	and
	
    ∫_1/2^1
    		(
    			𝗒 + √((1 - 𝗒) 𝗒) - 12)
    		(
    		2 - 𝗒^-1√(𝗒+2 𝗒√((1-𝗒 ) 𝗒)))  d𝗒=π-2/8.

	Combining the last two integrals, we obtain γ_1/2=1-π/4.
	This completes the proof.




 §.§ Fluctuations of the permutation height function



Let us now apply the second part of
<Ref> to the Grothendieck random permutations, and
obtain fluctuations of the height function
around its limit shape 𝗁^∘(𝗑,𝗒).
Throughout this subsection, we assume that the point (𝗑,𝗒)
belongs to the curved triangle 𝒞
where the height function 𝗁^∘ is not linear.

Recall that by H(x,y) we denote the random height function of the
Grothendieck random permutation 𝐰∈ S_n.
By (<ref>), we have
for all r∈ℝ:

    _𝐰(
    			H(⌊ n𝗑⌋,⌊ n𝗒⌋)
    			≤
    			n𝗁^∘+ r n^1/3)
    		
       =
    		_TASEP(
    			ξ̅_⌊ n(1-𝗑-𝗁^∘-r n^1/3) ⌋
    				(⌊ n𝗒⌋ )
    				≥
    				n(1-𝗒-𝗁^∘-r n^1/3)
    		)+o(1),

where we used the shorthand notation

    𝗁^∘
    	=
    	𝗁^∘(𝗑,𝗒)
    	=
    	1+
    	2/p√((1-p) x y)
    	-x+y/p
    	.

The term o(1) in (<ref>)
is due to the fact that we removed
shifts by (+1), and combined the integer parts. These
modifications are negligible because fluctuations live on the scale n^1/3.

By the fluctuation result for TASEP,
the probability in the right-hand side of
(<ref>)
has the following behavior:

    lim_n→∞_TASEP(
    			ξ̅_⌊ n(1-𝗑-𝗁^∘-r n^1/3) ⌋
    				(⌊ n𝗒⌋ )
    				≥
    				n(1-𝗒-𝗁^∘-r n^1/3)
    			)
       =
    		F_2( r( 1/𝗏_1(1-𝗑-𝗁^∘,𝗒)
    		+
    		1/𝗏_2(1-𝗑-𝗁^∘,𝗒)) ),

where F_2 is the Tracy–Widom GUE cumulative distribution function,
and the constants 𝗏_1,𝗏_2 are given by
(<ref>).
We have

    1/𝗏_1(𝗆,𝗍)
    	+
    	1/𝗏_2(𝗆,𝗍)=
    	(√(𝗍)-√(𝗆 p)) 𝗍^1/6/𝗆^1/3
    	(√(𝗍/p)-√(𝗆))^2/3
    	(√(𝗍 p)-√(𝗆))^2/3.

Let us denote the above expression with
𝗆=1-𝗑-𝗁^∘(𝗑,𝗒)
and 𝗍=𝗒 by

    𝗏(𝗑,𝗒)(√(𝗒)-√(p(1-𝗑-𝗁^∘(𝗑,𝗒))))
    	𝗒^1/6/(1-𝗑-𝗁^∘(𝗑,𝗒))^1/3
    	(√(𝗒/p)-√(1-𝗑-𝗁^∘(𝗑,𝗒)))^2/3
    	(√(𝗒 p)-√(1-𝗑-𝗁^∘(𝗑,𝗒)))^2/3,

where 𝗁^∘(𝗑,𝗒) is given in (<ref>).
While it is not evident from (<ref>),
inside 𝒞
the function 𝗏(𝗑,𝗒)
is symmetric in 𝗑,𝗒.

With the constant 𝗏(𝗑,𝗒), the convergence
in (<ref>)
implies the desired
Tracy–Widom GUE fluctuations of the height function
H(x,y).
This completes the proof of
<Ref>.



§ RANDOM PERMUTATIONS FROM NON-REDUCED PIPE DREAMS



Here we consider a different family of random permutations
which is
obtained by sampling a non-reduced pipe dream and not resolving any double crossings.
This corresponds to setting q=1 in <Ref>
from the Introduction.
For shorter notation, throughout this section we denote these random permutations
𝐰^(1) by 𝐰.



 §.§ Exact formula



Consider the staircase shape
_n= { (i,j) i+j≤ n }
(where i and j are the row and column indices, respectively)
and
place tiles
-2pt
    < g r a p h i c s >

or
-2pt
    < g r a p h i c s >

in each box of _n independently with probabilities p and 1-p, respectively.
Let the random permutation 𝐰∈ S_n
is obtained by following the pipes starting at (i,0), and ending at (0,𝐰_i^-1),
i=1,…,n.
For example, for the pipe dream in <Ref>, this procedure gives
w=241635.

Observe that each pipe is a random walk in the bulk of
_n, equipped with the mandatory turns by
90^∘ at the elbows at the diagonal boundary i+j=n.
We begin by deriving an exact formula for the probability
distribution of 𝐰_i^-1, the outgoing position
of the i-th pipe. For general p, this distribution
involves two Gauss hypergeometric functions _2F_1, which
for p=1/2 simplifies to a sum of binomial
coefficients. We do not use these exact formulas for
asymptotic analysis, but instead, in
<Ref> below, we obtain bounds on
the expected number of inversions using the random walk
approach.






Recall the Gauss hypergeometric function

    _2F_1(a,b;c;z) = ∑_k≥ 0 (a)_k (b)_k/(c_k)z^k/k!,

where (a)_k = a(a+1)⋯ (a+k-1). Set

    F_1(i,j)   
    		(1-p)p^i+j-2_2F_1(-i+1,-j+1;1; (1/p-1)^2)
    		
       = ∑_k=0^∞i-1kj-1k (1-p)^2k+1 p^i+j-2-2k;
    
    		F_2(s,n)     n (1-p)^2p^n+s-3_2F_1(-s+2,-n+1,2, (1/p-1)^2 )
    		
       = ∑_k=0^∞s-2knk+1 (1-p)^2k+2 p^n+s-3-2k.

Note that both hypergeometric series terminate after finitely many terms.
For p=1/2, these functions simplify to binomial coefficients:

    F_1(i,j)|_p=1/2
    	=2^-i-j+1i+j-2j-1,
    	    
    	F_2(s,n)|_p=1/2
    	=2^-s-n+1n+s-2n-1.



	
	We have for all i,j=1,…,n:
	
    (𝐰^-1_i=j) = F_1(i,j) +
    			p^n 1_i+j=n+1
    			+
    			F_2(i+j-n,n)1_i+j>n+1.



    We proceed by induction on i.
		For i=1, we have
		
    (𝐰^-1_1=j) =
        
        (1-p) p^j-1,     if  1≤ j<n;
    
    		p^n-1,     if  j=n,

    which coincides with the proposed formula. Now let i>1.
		Let X_i^(n) denote the exit position of the pipe started at (i,0),
		assuming that all squares on the diagonal i+j=n+1 are elbows.
		By considering where this pipe crosses the first row,
		we can represent
		X_i^(n)=X_i-1^(n-1) + Y,
		where Y depends on X_i-1^(n-1) and has the following
		conditional distribution
		given X_i-1^(n-1):
		
    (Y=m| X_i-1^(n-1)) =
    			
    				p,    m=0;
    
    				(1-p)^2  p^m-1,    1≤ m< n-X_i-1^(n-1);
    
    				(1-p)  p^n-X_i-1^(n-1)-1,    m=n-X_i-1^(n-1).

		Since 𝐰^-1_i=X_i^(n),
		we can write
    
    (X_i-1^(n-1)=j)· p+∑_r< j(X_i-1^(n-1)=r) ·(𝐰^-1_i =j)
    					
       
    					= 
    					(1-p)^2 p^j-r-1,     j<n;
    ∑_r< n(X_i-1^(n-1)=r)· (1-p)  p^n-r-1,      j=n.

		Using the induction hypothesis,
		we know the distribution of X_i-1^(n-1)
		in terms of F_1 and F_2.
		Therefore, it remains to verify the recursion
		(<ref>)
		for the answer (<ref>).
		This is easily done by grouping binomial coefficients,
		summing over r, and using the hockey-stick identity.





 §.§ Estimates from random walks



We are interested in the asymptotics of the (expected) number of inversions
(𝐰) in the random permutation 𝐰 obtained from the non-reduced pipe dream model.
First, we use the following known bound in terms of a
displacement (disarray)
dis(w) ∑_i=1^n |i-w_i|
=
∑_i=1^n|i-w_i^-1|:


	For any permutation w, we have
	 
    1/2dis(w) ≤(w) ≤dis(w).


Taking the expectation of (<ref>) and using linearity,
we see that it suffices to understand the asymptotic behavior
of 𝔼[|i-𝐰_i^-1|] for all i.
In principle, the exact distribution of 𝐰_i^-1
from <Ref>
should allow us to compute this expectation.
However, this is not straightforward.
Instead, we use the random walk interpretation of each individual
pipe:


	Fix ε>0 and let i<(1-ε)n. Then
	
    𝔼[|i-𝐰_i^-1|]
    		=
    		√(4i/πp/1-p) + O(1),     n→∞.



	Fix i. Let J_i,J_i-1,…,J_1,J_0
	be the (random) column coordinates of the positions of the i-th pipe
	in row i,i-1,…,1,0.
	In particular, J_i=0 and J_0=𝐰^-1_i.
	For example, in the pipe dream in <Ref>, left, we have
	J_3=0, J_2=3, J_1=3, and J_0=5 for i=3.

	We aim to upper bound the probability that
	the pipe reaches the diagonal i+j=n first time at row k_0, that is,
	J_k_0=n-k_0. Denote this event by R_k_0,
	and observe that (R_a)≥(R_b) for a<b.
	Before the pipe reaches the diagonal,
	the horizontal displacements J_k-J_k+1, k_0<k<i-1, are iid (independent
	identically distributed), and are distributed as
	
    ( {J_k-J_k+1=m}∩ R_k_0)=p1_m=0
    		+
    		(1-p)^2p^m-11_m≥ 1,     m∈ℤ_≥ 0,
    		   k_0<k<i-1.

	One readily checks that the
	expectation and variance of the random variable in the right-hand side are equal to 1
	and 2p/1-p, respectively.
	We can represent
	
    J_k_0=
    		J_k_0-J_k_0+1+J_i-1+∑_k=k_0+1^i-2( J_k-J_k+1).

	For the event R_k_0 to occur, the above sum must be
	equal to n-k_0=i-k_0+(n-i)>i-k_0+n ε.
	Since the expectations of the iid summands are equal to 1,
	by a standard large deviation estimate, this probability
	is upper bounded by e^-c_ε n for suitable
	c_ε(k_0)>0.[Note that if i-k_0≪ n, the actual bound is even stronger than this, but we do not need this precision.]
	By monotonicity, we can choose these constants such that c_ε(k_0)≥ C_ε c_ε(0)>0
	for all k_0.
	Taking the union over all k_0, we see that the probability
	that a pipe started at i ever reaches the diagonal
	is exponentially small.

	Therefore, J_0=𝐰^-1_i
	is close (with exponentially small error in probability)
	to a sum of i iid random variables
	J_k-J_k+1.
	By the Central Limit Theorem,
	this sum is approximately normal with mean i and variance
	2ip/(1-p). Subtracting i from 𝐰^-1_i, taking the absolute value,
	and using the fact that 𝔼( |Z| )=√(2/π)
	for standard normal Z, we obtain the result.


We can now bound the expected number of inversions in 𝐰 on the scale
n^3/2:

	
	Fix p∈[0,1).
	For every ε
		>0 and sufficiently large n, we have
		
    2/3√(π) (1-ε)  n^3/2√(p/1-p)≤𝔼[(𝐰)] ≤4/3√(π) (1+ε)  n^3/2√(p/1-p).



		For p=0, the permutation is identity with probability 1,
		so the bounds
		(<ref>) hold trivially.

    Let p>0 and let us calculate 𝔼[dis(𝐰)] to apply <Ref>.
		With <Ref> for i< (1-ε)^2/3n, we obtain
		for large n:
    
    𝔼[dis(𝐰)] =∑_i=1^n 𝔼[|𝐰_i^-1-i|] ≥∑_i=1^(1-ε)^2/3n2/√(π)√(p/1-p)√(i)≥2/√(π)√(p/1-p)2/3 n√(n) (1-ε),

    where the last inequality is a simple Riemann sum approximation. The upper bound follows similarly, this time extending the Riemann sum to n.


Numerical simulations suggest the following behavior of the
number of inversions:

	
	For any p∈[0,1), we have convergence in probability:
	
    lim_n→∞(𝐰)/n^3/2 = ϰ√(p/1-p).


Note that the bounds
2/(3√(π))≈ 0.376 and 4/(3√(π))≈ 0.752
in <Ref>
are at the same time bounds on ϰ.
Simulations show that ϰ is close to 0.5, but is not exactly equal to it.

The fact that the number of inversions lives on scale
n^3/2
implies a trivial permuton limit of 𝐰.
Denote
by id the
deterministic identity permuton
supported on the diagonal of [0,1]^2.


	
	For any fixed p∈ [0,1), the random permutations 𝐰∈ S_n converge in distribution
	to id as n→∞.


	For any τ∈ S_k and w∈ S_n, k< n, denote by
	t(τ,w)
	<cit.>
	the relative frequency of
	the pattern τ in w.
	This is simply the number of times the pattern τ appears in w,
	divided by nk.
	In particular, t( 21, w)=(w)/ n2.
	We have for any 2<ℓ < n
	<cit.>:
	
    t(21, w) = ∑_τ∈ S_ℓt(21, τ) t(τ, w).

	The term in the sum with τ=id vanishes since t(21, id)=0.
	For our random permutations
	𝐰∈ S_n, the left-hand side of (<ref>)
	converges to zero in probability by <Ref>
	and Markov inequality:
	
    (
    			(𝐰) ≥δ n^2)≤ C_δ n^-1/2→ 0,     n→∞

	for all δ>0.
	For a fixed ℓ, this implies that
	t(τ,𝐰)→0,
	n→∞,
	in probability
	for all idτ∈ S_ℓ, ℓ≥3. We see that all relative frequencies
	converge to zero except for t(id, 𝐰)→ 1.
	Therefore,
	by
	<cit.>,
	we get the result.


This completes the proof of <Ref> from the Introduction.



§ MAXIMAL PRINCIPAL SPECIALIZATIONS OF GROTHENDIECK POLYNOMIALS





 §.§ Grothendieck polynomials via bumpless pipe dreams



In this subsection, we outline another
combinatorial model for Schubert and Grothendieck polynomials
based on bumpless pipe dreams. Some of these definitions and results
go back to
<cit.>, and are described in detail in
<cit.> and <cit.>.

A bumpless pipe dream is a
tiling D of the n× n square
with six types of tiles:

    SE bump  -2pt
        < g r a p h i c s >
    ,   NW bump  -2pt
        < g r a p h i c s >
    ,   cross  -2pt
        < g r a p h i c s >
    ,   empty  -2pt
        < g r a p h i c s >
    ,   horizontal  -2pt
        < g r a p h i c s >
    ,   vertical  -2pt
        < g r a p h i c s >
    .

These tiles form a set of pipes labeled 1,…,n going
from the bottom boundary (left to right) to the right
boundary. See <Ref>, left, for an illustration.
We denote the set of all bumpless pipe dreams of size n by (n).


	
	Bumpless pipe dreams are the same as
	configurations
	of the six-vertex model in the n× n square
	with domain wall boundary conditions (see <cit.> or <cit.>).
	We refer to <cit.>
	or <cit.> for background and many properties
	on the six-vertex model. This object is also sometimes called
	the oscullating lattice paths model <cit.>.

	The same configurations can be identified with
	alternating sign matrices (ASMs) of size n,
	see <cit.> for a detailed exposition,
	and <cit.>, <cit.> for applications
	of the six-vertex model to the enumeration of ASMs.






	
	A permutation w(D) is associated to a bumpless pipe dream D
	by tracing the pipes from left to right, starting from the bottom boundary,
	and ignoring the second and subsequent crossings of each pair of pipes.
	See <Ref>, right, for a reading of the
	bumpless pipe dream in the left part of the figure
	which leads to w(D). The local rules for labeling the
	tiles in a bumpless pipe dream
	(which produce the permutation w(D))
	are given in
	<Ref>.
	Denote by (w)⊂(n) the set of bumpless pipe dreams
	whose associated permutation is w.





	A bumpless pipe dream is called reduced if no two pipes
	cross twice.
	Denote the set of reduced bumpless pipe dreams of size n by (n),
	and by (w)⊂(n) the subset
	whose associated permutation is w.


For a bumpless pipe dream D (reduced or not), denote by
(D) and (D) the set of NW bumps and empty tiles in D, respectively
(see (<ref>) for the notation).
For a tile in a bumpless pipe dream, we denote its
horizontal and vertical coordinates by (k,l), 1≤ k,l≤ n.
Here k increases from left to right, and l increases from top to bottom.

 
	For any permutation w∈ S_n, we have
	
    𝔊^β_w(x_1,…,x_n ) = β^-ℓ(w)∑_D ∈(w)∏_(k,l)∈(D)β x_k ∏_(k,l)∈(D) (1+β x_k).

	In particular, for β=0, we have
	for the Schubert polynomials:
	
    𝔖_w(x_1,…,x_n ) = ∑_D∈(w)∏_(k,l)∈(D) x_k.




 
	Recall the notation (<ref>).
	We have
	
    v_n(β)    =   ∑_D ∈(n)β^#(D)-ℓ(w(D)), 
       =∑_P ∈(n)β^#(P)-ℓ(w(P))(1+β)^#(P),

	where w(D) and w(P) are the
	permutations associated to the pipe dream D and bumpless pipe dream P, respectively.


	These identities follow from the formulas for Grothendieck polynomials in <Ref>.
For (<ref>), we immediately get

    v_n(β) = ∑_w 𝔊_w^β(1^n) = ∑_D ∈ PD(n)β^#(D)-ℓ(w(D)).

For (<ref>), we have

    v_n(β)= ∑_w 𝔊_w^β(1^n)    = ∑_P ∈ BPD(n)β^-ℓ(w(P))∑_D ∈(w)∏_(k,l)∈(D)β∏_(k,l)∈(D) (1+β )
       = ∑_P ∈ BPD(n)β^#(P)-ℓ(w(P))(1+β)^#(P).

This completes the proof.



Gao and Huang  <cit.> gave a bijection between reduced pipe dreams and reduced bumpless pipe dreams that preserves the corresponding weight of the diagrams in the corresponding expansions (<ref>) and (<ref>) of Schubert polynomials. This bijection was recently generalized to the case of pipe dreams and bumpless pipe dreams with marked NW-bumps and Grothendieck polynomials by Huang–Shimozono–Yu <cit.>.



 
By the product
formula for the number of alternating sign matrices of size n
<cit.>, <cit.>, we have
#(n) = ∏_k=0^n-1(3k+1)!/(n+k)!. Moreover,
since (n)⊂(n), we have

    u_n(0) ≤ v_n(0) ≤∏_k=0^n-1(3k+1)!/(n+k)!.

From the asymptotics of the number of AMSs of order n
(which is relatively straightforward using the product formula),
we conclude that

    lim sup_n→∞1/n^2log_2 u_n(0)   ≤ 0.37.

The asymptotic behavior of the number of bumpless pipe dreams
(six-vertex model configurations)
with periodic boundary is originally due to
Lieb <cit.>. The domain wall case
with general six-vertex parameters is treated
by Korepin–Zinn-Justin <cit.>.




 §.§ Asymptotics of Grothendieck random permutations via domino tilings



Consider a random bumpless pipe dream 𝐃∈(n)
with the distribution

    _2-ASM(𝐃=D) = 2^-n2 2^#(D),    
    	D∈(n).

Via the bijection with Alternating Sign Matrices
(<Ref>),
this distribution is equivalent to choosing each alternating
sign matrix of size n with probability
proportional to 2^#(number of -1's).
We call this distribution the 2-enumerated ASM model,
and it is equivalent (via a two-to-one bijection) to
the uniform probability distribution on
domino tilings of the Aztec diamond of size n-1.
This connection, as well as
the asymptotics of uniformly random domino tilings,
has been a subject of extensive research in
statistical mechanics and integrable probability,
see, for example,
<cit.>, <cit.>, and
<cit.>, <cit.>, <cit.>.

The results from the previous
<Ref> immediately imply:

	
	The Grothendieck random permutation 𝐰∈ S_n
	with the parameter p=1/2
	(see <Ref> for the definition)
	coincides with the permutation w(𝐃)
	associated
	(as in <Ref>)
	to the
	random bumpless pipe dream 𝐃∈(n)
	with the distribution (<ref>).



	
	For general p, an analog of <Ref>
	would involve a more complicated weighting of bumpless pipe dreams
	(six-vertex configurations). This weighting is still in the
	free-fermion family of six-vertex weights.
	At the level of domino tilings, the p-weighting corresponds to a
	deformation of the uniform model
	in which vertical and horizontal dominoes have different weights.
	This model is also well-studied,
	for example, see <cit.>.
	For simplicity, here we focus only on the case p=1/2.


Our asymptotic results for
Grothendieck random permutations (<Ref>)
readily
translate into results about the behavior of the permutations
w(𝐃) coming from the 2-enumerated ASM model.
On the other hand, the asymptotic behavior of the bumpless pipe dream 𝐃
in various regimes
has also been extensively studied,
and some of these results can be
brought back to Grothendieck random permutations.
Let us present just one example leading to asymptotically Gaussian fluctuations:


	
	Let 𝐰∈ S_n be the Grothendieck random permutation with p=1/2.
	Let 𝐰_n be the image of n under 𝐰.
	We have
	
    𝐰_n-n/2/√(n/4)d⟶𝒩(0,1),    
    		n→∞.

	Here 𝒩(0,1) is the standard normal distribution.


	Let
	𝐥_n be the position of the single NW bump in the
	rightmost column of 𝐃.  In
	<Ref>, we have
	𝐥_n=5.
	By <Ref>, 𝐰_n has the
	same distribution as 𝐥_n.
	In the language of domino tilings
	(see the map in, e.g., <cit.>), 𝐥_n is the position of the
	first particle in the particle process associated to the domino tiling.
	By
	<cit.>,
	𝐥_n satisfies a central limit theorem as in
	(<ref>). Note that our standard deviation
	√(n/4) matches the normalization in <cit.>
	as the latter work deals with GUE random matrices
	whose diagonal elements have variance 2.
	This completes the proof.



	
	<Ref> can also be proven directly by looking at the trajectory of the
	pipe of color n in the random pipe inside the staircase shape
	(<Ref>), as this trajectory is a random walk.




 §.§ Exact formulas for elementary layered permutations



We are now in a position to discuss principal specializations
of Grothendieck polynomials on layered permutations.
We begin with a few exact formulas, for which we need some notation.

Let 𝖢_n1/n+12nn denote the
nth Catalan number which counts the number of
Dyck paths of size n. Let _n denote the n-th
little Schröder number
<cit.> and
_n the n-th large Schröder number
<cit.>. It is
known that _n=2_n for n≥ 1.

The Narayana polynomial is defined as

    ℒ_n(x) ∑_d ∈Dyck(n) x^v(d),

where the sum is over Dyck paths of size n and v(d) is the number of valleys of the path d. The coefficients of ℒ_n(x) are the Narayana numbers N(n,k)=1/knknk-1 <cit.>. Note that 𝖢_n=ℒ_n(1) and s_n=ℒ_n(2). The polynomials ℒ_n(x) have the following generating function:

    ∑_n≥ 0 y^n ℒ_n(x) = 1-y(1+x)-√((1-y(1+x))^2-4xy^2)/2y.



Consider the following
elementary layered permutation
(see also (<ref>) for notation):

    w_0(k;n)id_k × w_0(n) = (1,…,k, k+n,k+n-1,…,k+1)
    	 ∈ S_k+n.

Recall the notation Υ_w(β)
(<ref>).
A
formula due to Fomin and Kirilov <cit.> relates the
Schubert principal specialization Υ_w_0(a;b)(0)
to Proctor's formula <cit.> counting bounded
plane partitions of staircase shape:


	
	For nonnegative integers k and n, we have
    
    Υ_w_0(k;n)(0) = 𝔖_w_0(k;n)(1,…,1)  = [𝖢_n-2+i+j]_i,j=1^k  = ∏_1≤ i<j≤ n2k+i+j-1/i+j-1.



In <cit.>, an analogous formula for
the β=1
Grothendieck polynomials is proven:


	
	For nonnegative integers k and n, we have
	
    Υ_w_0(k;n)(1) = 𝔊^β=1_w_0(k;n)(1,…,1) = 2^-k2[_n-2+i+j]_i,j=1^k
    		= 2^-k+12[_n-2+i+j]_i,j=1^k.



A general determinantal formula for all β may be derived
from <cit.>:
  
	For nonnegative integers k and n, we have

    Υ_w_0(k;n)(β)  =  (1+β)^-k2[ℒ_n+i+j-2(1+β)]_i,j=1^k.



The determinants
in
<Ref>
can be efficiently computed via the Dodgson condensation (the Desnanot–Jacobi identity).
We use it to get the numerical data for β=1
presented in
<Ref>.



 §.§ Asymptotically maximal specializations for layered permutations



Here we prove that principal
specializations of β=1
Grothendieck polynomials on layered permutations
(see (<ref>) for the definition)
attain the asymptotically largest
value. That is, we prove <Ref> from the Introduction.


Consider the compositions b^(n) = (…, b_2, b_1) of n
such that

    b_1 + … + b_i-1 + (2 + √(2))b_i ≤ n  for  i = 1,2,…,
     while b_1 + … + b_i-1≤ n - 4,

and b^(n) has o(n) parts.
Then for the layered permutations w(b^(n)) we have

    lim_n →∞1/n^2log_2
    Υ_w(b^(n)) (1)
    = 1/2,

which is asymptotically maximal.



	In particular, one can take compositions to be geometric
	b_i ∼ (1 - α) α^i - 1 n, for any α∈ [1/√(2), 1).


In the rest of this <Ref>,
we prove <Ref>. First, observe that the Grothendieck
specializations enjoy the following basic factorization property:

    Let u ∈ S_k, w ∈ S_n. We have
    
    Υ_u × w(β) = Υ_u(β) ·Υ_id_k × w(β).




	This follows from properties of pipe dreams
	corresponding to permutations
	of the form u × w.
	See <cit.> or <cit.>.


Let us denote (recall the notation (<ref>))

    F(k, n) Υ_w_0(k;n)(1),      G(k,n)   2^k2· F(k,n)=
    	[_n-2+i+j]_i,j=1^k,

where the last equality follows from <Ref>.



An important step in obtaining the asymptotics of Υ_w(1) for layered permutations is to understand the behavior of the function F(k,n). A similar analysis <cit.> was possible in the Schubert case (β = 0) thanks to an explicit product formula (<Ref>). For other β's, we have determinantal (but not product) formulas (<Ref> and <Ref>). However, in the case β=1 we are able to asymptotically analyze these determinants via the correspondence with 2-enumerated ASMs.


Let
k ∈ (n/√(2), n]. We have

    log_2 F(k, n - k) = n^2/2 - k^2/2 - O(n),       n →∞.


In (<ref>) and throughout the proofs below in this subsection,
we assume that the constants in the O(·) notation are nonnegative.

	Let 𝐃∈ PD(n) be the
	random bumpless pipe dream with the
	distribution (<ref>).
	Then we have
	
    F(k,n-k)=2^ n2(w(𝐃)=w_0(k;n-k)).

	Let us first consider a different event,
	
    R_k,n-k{restriction of w(𝐃) to the last n-k entries equals
    				(n,n-1,…,n-k+1)}.

	That is, under R_k,n-k, the first k entries can be an arbitrary
	permutation of {1,…,k }, while the last n-k entries are in the fully reversed order.
	Geometrically, event (<ref>)
	means that the bumpless pipe dream 𝐃
	develops a right triangle of side n-k in the lower right corner
	whose diagonal boundary is made
	of the SE bumps -2pt
    < g r a p h i c s >
, and
	the rest of the triangle
	entirely consists of crosses -2pt
    < g r a p h i c s >
.
	See <Ref> for an illustration.



	The 2-enumerated ASM model is equivalent to uniform domino tilings of the Aztec diamond,
	which develops an arctic circle phenomenon <cit.>. In terms of bumpless pipe dreams, this means that the pipe configuration outside of the circle inscribed in the n× n square is nonrandom (frozen). The four frozen phases adjacent to the four vertices of the square are different, but the one in the lower right corner consists entirely of crosses.
	By the variational principle for domino tilings
	<cit.>,
	as long as our triangle of side n-k is outside the arctic circle
	(which is equivalent to n/√(2)<k≤ n), the condition
	in R_k,n-k does not lower the asymptotics of the probability on the
	scale e^-cn^2.
	Moreover,
	
    (R_k,n-k)=1-e^-O(n),      n→∞.

	Indeed, the complementary event to R_k,n-k
	is that there is at least one non-cross in the lower right triangle of side n-k.
	By <cit.>,
	this event has probability e^-O(n).

	Returning to F(k,n-k),
	using the factorization property
	(<Ref>) and the fact that v_k(1)=2^k2, we have
	
    2^ n2-O(n)∼∑_w∈ S_kΥ_w × w_0(n-k)(1)
    		=
    		∑_w∈ S_kΥ_w(1) · F(k,n-k)=
    		2^ k2· F(k,n-k).

	This implies the desired asymptotic equivalence.


	
	The recent work <cit.>
	provides asymptotics of
	log_2 F(k, n - k) for all k, not necessarily in the range k>n/√(2).
	For k<n/√(2), the leading term
	is strictly smaller than n^2/2,
	which agrees with the variational principle
	<cit.>.
	We do not need the regime k<n/√(2) for <Ref>.




Let us denote k_0  n and

    k_i  n - b_1 - … - b_i,
       i ≥ 1,

so that b_i = k_i-1 - k_i.
We have by <Ref> and (<ref>) that

    Υ_w(…, b_2, b_1)(1)    = Υ_w(…, b_3, b_2)(1) · F(n-b_1,b_1)  
       = ∏_i ≥ 1 F(n - b_1 - … - b_i, b_i) = ∏_i ≥ 1 F(k_i, k_i-1 - k_i).

From the inequalities on the b_i's, we get

    k_i/k_i-1 = n - b_1 - … - b_i/n - b_1 - … - b_i-1∈ (1/√(2), 1).

Hence by <Ref>, we have

    log_2 F(k_i, k_i-1 - k_i) = k_i-1^2/2 - k_i^2/2 - O(k_i-1).

Therefore,

    1/n^2log_2 Υ_w(…, b_2, b_1)(1,…, 1)    = ∑_i ≥ 11/n^2log_2 F(k_i, k_i-1 - k_i) 
       = ∑_i ≥ 11/n^2(k_i-1^2/2 - k_i^2/2 - O(k_i-1) ) 
    
       = 1/2 - 1/n^2 O(∑_i ≥ 1 k_i-1).


Since k_i-1 = O(n) and there are o(n) elements in our composition, we have

    ∑_i ≥ 1 k_i = o(n^2),  and so 1/n^2 O(∑_i ≥ 1 k_i-1) = o(1),








which completes the proof of <Ref>.






	
	Recall the notation
	(<ref>),
	(<ref>).
	<Ref> establishes that on layered permutations,
	the Grothendieck specializations Υ_w(1)
	reach their asymptotic maximum: u_n'(1)∼ 2^ n2.
	However, it is not clear whether
	the absolute maximum, u_n(1),
	is attained on layered permutations.
	Dennin
	<cit.> verified that for n≤ 9, one has
	u_n(1)=u_n'(1).

	We present numerical data of
	optimal (i.e., achieving the exact, non-asymptotic maximum in u_n'(1))
	layered
	permutations for n≤ 500 in
	<Ref>.
	We also comment on why these optimal layered permutations
	have parts whose behavior does not fall under the assumptions of <Ref>.




 §.§ Bounds on the constant for Grothendieck specializations with general β



In <cit.>, the authors conjectured that for β fixed there exists a limit

    c(β)lim_n→∞1/n^2log_2 υ_n(β).

If the limit exists, it must satisfy
for β≥ 0:

    1/4log_2(2+β) ≤ c(β) ≤1/2log_2(2+β),

see <cit.>.
The limit
(<ref>)
exists when β=1, see (<ref>),
and we have c(1)=1/2.
Here we
improve the bounds (<ref>) in certain regimes:



 
	If the limit c(β) (<ref>) exists, then it must satisfy the following bounds.
	For 0 < β≤ 1, we have
	
    1/4max{log_2(2+β), 2log_2(1+β)}≤ c(β) ≤1/2min{log_2(2+β), log_2(1+1/β)}.

For β≥ 1, we have

    1/4max{log_2(2+β), 2log_2(1+1/β)}≤ c(β) ≤1/2log_2(1+β).

				See <Ref> for an illustration of the bounds.


For 0<β≤ 1 and
D∈(n), we can bound 1 ≤β^-ℓ(w(D))≤β^-n2.
Thus by (<ref>), we have

    ∑_D ∈(n)β^#(D)≤ v_n(β) ≤β^-n2∑_D ∈(n)β^#(D),

which leads to
(1+β)^n2≤ v_n(β) ≤(1+β/β)^n2.
Taking the logarithms and comparing with (<ref>) gives the result.

For β≥ 1 and D∈(n), let us estimate
β^-n2≤β^-ℓ(w(D))≤ 1. Thus by (<ref>), we have

    β^-n2∑_D ∈(n)β^#(D)≤ v_n(β) ≤∑_D ∈(n)β^#(D),

yielding
(1+β/β)^n2≤ v_n(β) ≤ (1+β)^n2,
which leads to the
desired bounds.








§ ADDITIONAL REMARKS AND OPEN PROBLEMS





 §.§ Optimal permutations achieving the exact maxima u_n(1) and u_n(β)


This project was motivated by Stanley's original question
from <cit.> on determining the family of
permutations w for which the Schubert specialization 𝔖_w(1^n) is
asymptotically maximal. This question as well as the
asymptotic value remain widely open (see <cit.> for data up to n≤ 10).
Dennin <cit.> extended this
question to Grothendieck specializations 𝔊_w^β=1(1^n),
but one can ask the same question for general β.
Presumably, answering this
question would give rise to a family of optimal permutations
w^β, which for β=0 would answer the original
Schubert specialization question.

Among the candidates for
such a family w^β are the layered permutations
<cit.>,
<cit.>
(see
<Ref> for numerical evidence for β=1).
When β=1,
Dennin verified that
layered permutations indeed achieve the maximum in u_n(1) for n≤ 9.
Our results of
<Ref> suggest that layered
permutations are indeed good candidates
as they at least attain the asymptotic maximum in u_n(1).

To obtain the asymptotics of Grothendieck specializations
on layered permutations for general β, one
has to study the asymptotics of the following Hankel
determinants:

    F(k,n,β)(1+β)^-k2[ℒ_n+i+j-2(1+β)]_i,j=1^k,

where ℒ_n(x) is the Narayana polynomial, see (<ref>)
and <Ref>.
For β=0, these determinants admit a product formula
and were analyzed in <cit.>.
We could access the special case β=1 due to the connection to
2-enumerated ASMs and domino tilings.
It would be interesting to find a deformation of the
domino tiling model related to the determinants (<ref>).



 §.§ Typical vs optimal permutations


One of our main results, <Ref>,
concerns “typical” Grothendieck
permutations.
By comparing the
permutation matrices in <Ref>, center,
and <Ref>, left, we see that
typical permutations are far from layered.
However, the
Shannon–McMillan–Breiman theorem (e.g., see
<cit.>)
suggests that probabilities of typical Grothendieck random permutations
also achieve the asymptotic maximum.
For our permutations, this is not yet a rigorous statement,
as one has to prove the asymptotic uniformity of the probability weights
of typical Grothendieck permutations.



 §.§ Local structure of Grothendieck permutations


What is the local structure of the Grothendieck random permutations?
In <Ref>, left and center, one
can observe a certain local layered structure.
Understanding this local structure goes beyond the permuton convergence.
Can the local structure be
understood in terms of descent statistics, major index, or
pattern statistics other than the overall pattern counts?
For instance, layered permutations are the
ones maximizing the packing density of the
pattern 132 <cit.>.
Potential tools
for this analysis may be found in
<cit.>. Alternatively,
one could translate the local behavior of random domino
tilings of the Aztec diamond <cit.>
to the local behavior of Grothendieck permutations.



 §.§ Random and maximal permutations for domain wall six-vertex model


The Bumpless Pipe Dreams provide a surprising connection
of Schubert calculus and K-theory to
the six-vertex model, one of the central objects in
statistical mechanics.
The analysis of typical and maximal permutations for Schubert
and Grothendieck specializations prompts the analogous questions
about the six-vertex model.

Namely, consider the six-vertex
model with
domain wall boundary conditions and
general Boltzmann weights a,b,c (for the 2-enumerated
ASMs, we have a=b=1 and c=√(2)).
Represent a random configuration of the six-vertex model
as a collection of pipes, and associate to it a random
permutation 𝐰 by resolving the intersections
as in
<Ref>.
What is the behavior of 𝔼[(𝐰)] and
how does it vary depending on the weights of the six-vertex
model? Do the permutations 𝐰 have a permuton limit?



 §.§ q-analogues and generalized principal specialization


Zhang <cit.> extended the analysis in <cit.> to the generalized principal Schubert specialization 𝔖_w(1,q,q^2,…), when q is a  root of unity. The analogue of layered permutations is the multi-layered ones <cit.>. It would be interesting to extend our results to the Grothendieck specializations
	𝔊^β_w(1,q,q^2,…) at a root of unity.
	However, having q complex makes connections to probabilistic models less clear.







§ DATA FOR MAXIMAL SPECIALIZATIONS ON LAYERED PERMUTATIONS





 §.§ Dodgson condensation and data


Principal specializations Υ_w(b)(β) of
Grothendieck polynomials with general β on layered permutations
(see <Ref> for notation)
can be maximized using dynamic programming, similarly to how this is done
for Schubert specializations in <cit.>.
To implement it, one needs an efficient way to compute the
determinants
D_β(n,k)[ℒ_n+i+j-2(1+β)]_i,j=1^k
from
<Ref>.
A recursion
follows from
the Dodgson condensation
(Desnanot–Jacobi identity)
<cit.>, <cit.>:


	For k,n∈ℤ_≥0,
	we have

    D_β(n,k)  = 1/D_β(n+2,k-2)(D_β(n+2,k-1)D_β(n,k-1) - D_β(n+1,k-1)^2),

with initial conditions D_β(n,0)=1, D_β(n,1)=ℒ_n(1+β), and D_β(0,k)=(1+β)^k2.


	Let M(n,k)=[ℒ_n+i+j-2(1+β)]_i,j=1^k, so
	D_β(n,k)= M(n,k). Clearly,
	
    D_β(n,0)=1,    
    		D_β(n,1)=ℒ_n(1+β).

	For n=0, the
	permutation w_0(0;k)=12⋯ k is the identity, and so
	𝔊^β_w_0(0;k)=1. By
	<Ref>, we have

    1 = Υ_w_0(n,k)(β) = (1+β)^-k2[ℒ_i+j-2(1+β)]_i,j=1^k.

Thus, D_β(0,k)= M(0,k)=(1+β)^k2.

For 1≤ i,j,p,q≤ k, let M_i,j^p,q denote the matrix obtained from M
by deleting rows i and j and columns p and q.
Similarly, let M_i^p be obtained from M by deleting row i and column p.
By the Dodgson condensation, we have

    (M(n,k))(M(n,k)^1,k_1,k)  = (M(n,k)^1_1)(M(n,k)^k_k) - (M(n,k)^k_1)(M(n,k)^1_k).

Observe that

    M(n,k)^1,k_1,k   = M(n+2,k-2),   M(n,k)^1_1 = M(n+2,k-1),
       M(n,k)^k_k   =M(n,k-1),   M(n,k)^k_1=M(n,k)^1_k=M(n+1,k-1).

This implies the result.


The implementation of the recursion for D_β(n,k) combined with dynamic programming
yields the data in <Ref>.
This data agrees with the asymptotic behavior f(n)→1/2
obtained in <Ref>.







 §.§ Optimal layered permutations and asymptotic maximum



Observe that the optimal layered permutations given in
<Ref> do not fall into the
description of sequences given in <Ref>.
For example, for n=700, the numerically optimal layered permutation
has parts b^(700)_opt=(1, 3, 5, 9, 14, 21, 33, 49, 73, 107, 157, 228),
whereas a sequence b^(700) from <Ref>
would satisfy the bound b_1=n-k ≤ 700/(2 + √(2)) ≈
205 on its largest part.
The parts of b^(700)_opt are suspiciously close to
the Fibonacci sequence (see <Ref>).
Let us explain this discrepancy.

Optimal
layered permutations can be constructed inductively via the
optimization

    max_b_1 + b_2 + ⋯ = nΥ_w(…, b_2, b_1)(1) =
    max_kmax_b_2 + b_3+⋯ = k{Υ_w(…,b_3, b_2)(1) · F(k, n-k)
    },

where b_1 = n-k.
Assume that max_b_2 +b_3+ ⋯ = kΥ_w(…,
b_3,b_2)(1) = 2^ k2· g(k), where g(k) =
2^-o(k^2). To extract the first part b_1, we thus need to
find

    max_k
    	{ 2^ k2· g(k) · F(k, n-k) }
    	=
    	max_k {g(k)· G(k,n-k)},

see (<ref>) for the notation.
On the other hand, in the proof of <Ref> (in particular, see
<Ref>),
we estimated the asymptotic behavior of
F(k,n-k), and
found that the coefficient
by n^2 in log_2 F(k,n-k) does not depend on k as
long as k>n/√(2).
These two problems lead to different choices of k.
See also
the
illustration in <Ref>.





The discrepancy between optimal layered permutations and the ones appearing in
<Ref> should be of order o(n) in the large n limit. Indeed,
assume that k/n<1/√(2). Then, as in the proof of
<Ref> (see also <Ref>),
one can show that log_2 F(k,n-k)∼α n^2-k^2/2-O(n) for some
α<1/2, so
2^ k2 g(k)  F(k, n-k)∼ 2^α n^2-o(n^2),
which is not maximal.
We believe that the discrepancy of the part sizes
in optimal layered permutations is of order
o(n) as n→∞.



§ LIMIT SHAPE AND FLUCTUATIONS IN TASEP FROM CONTOUR INTEGRALS



In this section we compute the constants
𝖼,𝗏_1,𝗏_2
from <Ref>, and
show how to establish
<Ref> by an asymptotic analysis of the correlation kernel.
We omit tedious but standard
estimates,
as they are
very similar to those in <cit.>
or <cit.>.



 §.§ Fredholm determinant



Recall that ξ̅_m(t)=ξ_m(t)-ξ_m(0) is the displacement of the m-th particle
at time t.
By <Ref>,
we have

    _TASEP(ξ̅_m(t)≥ u)=
    	(λ_m≥ u),

where λ_m is a point of the determinantal point process
X(λ)={λ_j+m-j }_j=1^m⊂ℤ_≥0 with the correlation kernel
(<ref>):

    K(u_1,u_2)=1/(2π𝐢)^2dz  dw/z-ww^u_2/z^u_1+1( z-p/w-p)^m( 1-w/1-z)^t,
    	    
    	u_1,u_2∈ℤ_≥0,

where the contours are positively oriented simple closed curves satisfying
p<|w|<|z|<1.

	
	The probability
	in the right-hand side of
	(<ref>) is equal to the Fredholm determinant
	of the kernel K on {0,1,…,u-1}:
	
    (λ_m≥ u)=
    		[ Id-K ]_≤ u-1
    		1+∑_ℓ=1^∞(-1)^ℓ/ℓ!∑_i_1,…,i_ℓ=0^u-1[ K(i_a,i_b) ]_a,b=1^ℓ,

	where Id is the identity operator.


	The event {λ_m≥ u} means that in the determinantal
	point process X(λ), no particles
	are located in {0,1,…,u-1 }.
	It is a well-known property of determinantal point processes that this
	probability is given by a Fredholm determinant, see e.g.
	<cit.>.
	This statement can be traced back to the inclusion–exclusion principle
	(e.g., see <cit.>).
	Note that the sum over ℓ in (<ref>) is finite, as
	for u>ℓ, the ℓ×ℓ determinant must have identical rows.
	Further details on Fredholm determinants may be found in
	<cit.> or <cit.>.


Let us assume that
m=⌊ L𝗆⌋,
t=⌊ L𝗍⌋,
u=⌊ L𝗎⌋,
where L→∞. By <Ref>,
it suffices to consider the case 𝗍≥𝗆/p,
which we assume throughout the present
<Ref>.



 §.§ Steepest descent and law of large numbers



The asymptotic analysis of K(u_1,u_2) is done by
steepest descent (see, e.g., <cit.> for an accessible introduction).
The integrand in (<ref>) has the form

    1/z-ww^u_2/z^u_1+1( z-p/w-p)^m( 1-w/1-z)^t
    	=exp{ L (
    	S(z;u_1/L)
    	-S(w;u_2/L)
    	)
    	+O(1)
    	}/z(z-w),

where

    S(z;𝗎)
    	-𝗎log z+
    	𝗆log(z-p)
    	-𝗍log(1-z).

The term O(1)
arises from dropping integer parts in m and t.
It is
negligible in the limit because the determinantal process
will be scaled from discrete to continuous space.

The point λ_m is at the left edge of the determinantal process.
Therefore, to catch its asymptotic location 𝗎=𝖼(𝗆,𝗍),
we need to find a double critical point of the function
z↦ S(z;𝗎).


	
	Let us explain the need for the double critical point
	in more detail.
	By (<ref>), we have
	K(u,u)=(u∈ X(λ)), which is the density
	of the random point configuration X(λ).
	The equation S'(z;𝗎)=0 for single critical points
	is quadratic, and its discriminant is
	a function of p and our parameters (𝗆,𝗍,𝗎).
	Fox fixed 𝗆 (viewed as a parameter of the determinantal
	process and not the scaled index on the last part λ_m),
	there are three possibilities depending on (𝗍,𝗎):
	
		
  * If the discriminant is negative,
			then
			K(u,u) converges to a value strictly between 0 and 1. Therefore,
			at the scaled time 𝗍 around the scaled location
			𝗎 there is a random configuration of points from X(λ)
			of density strictly between zero and one.
		
  * If the discriminant is positive, then
			K(u,u) converges to 0 or 1,
			and around 𝗎 there is either a densely packed, or an empty
			region of points from X(λ).
		
  * The case of zero discriminant (double critical points)
			is at the transition between the two cases, and
			thus
			corresponds to the edge of the random configuration X(λ).
	


Solving the double critical point equations S'(z;𝗎)=S”(z;𝗎)=0 in
z,𝗎, we find

    z_cr=
    	√(p𝗍)-√(𝗆)/√(𝗍/p)-√(𝗆)
    	,    𝗎_cr=
    	(√(p𝗍)-√(𝗆))^2/1-p.

Here out of two solutions to quadratic equations, we picked
the one with the smaller 𝗎_cr which corresponds to the left edge of X(λ).
Note that the solution 𝗎_cr in (<ref>) is the same as
the constant 𝖼(𝗆,𝗍)
from <Ref>, which is not a coincidence.

Let us deform the integration contours in
the kernel (<ref>) to pass near the double critical point
z_cr. This point lies inside the interval (0,p), while
the original contours satisfy p<|w|<|z|<1.
Due to high multiplicity poles in the integrand,
the w contour should not be deformed through p, and
the z contour should not be deformed through 0 and 1.

We deform the contours as follows. First,
deform w such that the new w contour goes around p and passes to the right of z_cr.
Then, deform z through w, such that
the new z contour goes around 0 and passes to the left of z_cr.
Dragging z through w picks up a residue at z=w, which needs to be integrated over the new w contour.
However, taking the residue at z=w eliminated the pole at w=p, so the
new single integral is zero.
This implies that the contour deformation does not produce extra terms in the kernel.
See <Ref>
for an illustration.



To arrive at the law of large numbers λ_m/L →𝖼(𝗆,𝗍)
(recall that this is the same as 𝗎_cr),
it remains to show that changing 𝖼
by ε or (-ε) makes the Fredholm determinant (<ref>) go to 0 or 1,
respectively. To show convergence to 1, it suffices to
estimate |K(u_1,u_2)|≤ e^-δ L for
u_1,u_2≤ L (𝖼-ε), which follows from the steepest descent analysis.
To show convergence to 0,
it suffices to look at the density function K(u,u) for u around L(𝖼+ε).
In the limit, this density
is strictly positive, and so the event
λ_m≥ L (𝖼+ε) has probability tending to zero.
We omit further details.



 §.§ Asymptotic fluctuations



Consider the probabilities of fluctuations as in <Ref>:

    _TASEP( ξ̅_⌊ L𝗆- L^1/3α𝗏_1⌋
    		(⌊ L𝗍⌋ )≥
    	L𝖼 -
    	L^1/3β𝗏_2
    )= [
    		Id
    		-
    		K^L𝗍,
    		L𝗆 - L^1/3α𝗏_1]_≤ L𝖼-
    	L^1/3β𝗏_2 +o(1).

Assume that 𝗆 and 𝗍 are fixed
(recall that 𝖼,𝗏_1,𝗏_2 depend on them).
In the right-hand side of
(<ref>),
we dropped integer parts as
they do not affect the limit to continuous space.

One can show that in the
series
(<ref>)
for the
Fredholm determinant (<ref>),
terms where
i_a< L𝖼-s L^1/3 for at least one a=1,…,ℓ
(and s sufficiently large but finite), are negligible in the limit.
Thus,
it suffices to consider
K̃(ũ_1,ũ_2)
K^L𝗍,
L𝗆 - L^1/3α𝗏_1
(L𝖼+ũ_1 L^1/3,L𝖼+ũ_2 L^1/3),
where ũ_1,ũ_2∈ℝ.
This computation will also produce
the normalization
constants 𝗏_1 and 𝗏_2
for the Tracy–Widom GUE fluctuations.

In K̃,
deform the contours to pass near the double critical point
z_cr=z_cr(𝗆,𝗍) given by (<ref>).
The contribution outside a small neighborhood of z_cr vanishes in the limit.
In the neighborhood,
make a change of variables

    z=z_cr+L^-1/3z̃/ϱ,
    	    
    	w=z_cr+L^-1/3w̃/ϱ,

where ϱ>0 is to be determined.
The new contour z̃ goes from
e^-2π𝐢/3∞ to -1 to
e^2π𝐢/3∞,
and w̃ goes from
e^-π𝐢/3∞ to 0 to
e^π𝐢/3∞.
The function
(<ref>)
is Taylor expanded in L as

    L( S(z;𝖼+L^-2/3ũ_1 )-S(z_cr;𝖼+L^-2/3ũ_1 ) )
    		
       =
    		-z̃^3/3ϱ^3p^3/2/√(𝗆𝗍)(√(𝗍/p)-√(𝗆))^4/(1-p)^3 z_cr
    		+
    		z̃/ϱ z_cr( -ũ_1
    			+
    			α𝗏_1(√(p𝗍/𝗆)-1)/1-p).

Let us pick ϱ>0 such that the coefficient
by z̃^3 is (-1/3),
that is,

    ϱ=
    	√(p)/(1-p) z_cr(√(p𝗍)-√(𝗆))^2/3
    	(√(𝗍/p)-√(𝗆))^2/3/(𝗆𝗍)^1/6.

We also have
in the integrand:

    dz  dw/z(z-w)∼ -L^-1/3/ϱ z_crdz̃ dw̃/w̃- z̃.

Adding the difference
S(z_cr;𝖼+L^-2/3ũ_2 )-S(z_cr;𝖼+L^-2/3ũ_1 )
inside the exponent in the integrand (<ref>)
does not change the correlation kernel, since it is a
gauge transformation of the form
K̃(ũ_1,ũ_2)↦K̃(ũ_1,ũ_2)f(ũ_2)/f(ũ_1),
where f(·) is a nonvanishing function.
Thus, we have

    K̃(ũ_1,ũ_2)f(ũ_2)/f(ũ_1)∼
    	-L^-1/3/ϱ z_cr∫_e^-2π𝐢/3∞^e^2π𝐢/3∞
    	dz̃∫_e^π𝐢/3∞^e^-π𝐢/3∞
    	dw̃ exp[ w̃^3/3
    			-
    			z̃^3/3
    			+
    			z̃ U_1
    			-
    			w̃ U_2
    	]/w̃-z̃,

where

    U_1,21/ϱ z_cr(
    	-ũ_1,2
    		+
    		α𝗏_1(√(p𝗍/𝗆)-1)/1-p)
    	.

The integral in (<ref>) together with the minus sign from the prefactor
is equal to the celebrated Airy_2 kernel
𝖠(U_1,U_2)
(e.g., see <cit.>).
The Tracy–Widom GUE cumulative distribution function
is the Fredholm determinant of the Airy_2 kernel, that is,
F_2(r)=[Id-𝖠]_≥ r, r∈ℝ.

The additional prefactor (L^1/3ϱ z_cr)^-1
in (<ref>)
combined with the sums in
the pre-limit Fredholm determinant (<ref>)
turns them into Riemann integrals of the Airy_2 kernel.
The normalization by
ϱ z_cr
is absorbed by
a change of variables:

    1/L^1/3ϱ z_cr∑_i=⌊ L^1/3ũ⌋≤ -L^1/3β𝗏_2
    	g(ũ)
    	∼1/ϱ z_cr∫_-∞^-β𝗏_2
    	g(ũ) dũ=
    	∫_1/ϱ z_cr(
    		β𝗏_2
    		+
    		α𝗏_1(√(p𝗍/𝗆)-1)/1-p)
    	^+∞𝗀(U) dU.

Here we incorporated the bound
L𝖼- L^1/3β𝗏_2 from (<ref>)
into the bound for i, and shifted
all indices by L𝖼.
The function g(ũ) represents
terms like K(·, i_a)K(i_a,·) in the
Fredholm determinant (<ref>)
arising when expanding the individual ℓ×ℓ
determinants as sums over permutations.
Here ũ and i_a are related as
i_a=⌊ L^1/3ũ⌋.
We also set 𝗀(U)=𝗀(ũ).

We see that
the following
convergence of Fredholm determinants holds:

    lim_L→∞[
    		Id
    		-
    		K^L𝗍,
    		L𝗆 - L^1/3α𝗏]_≤ L𝖼
    	=
    	[
    		Id
    		-
    		𝖠]_≥ r,

where

    r=
    	1/ϱ z_cr(
    		β𝗏_2
    		+
    		α𝗏_1(√(p𝗍/𝗆)-1)/1-p).

We see that setting

    𝗏_1=
    		𝗏_1
    		(𝗆,𝗍)
    		   ϱ z_cr(1-p)/√(p𝗍/𝗆)-1
    		=
    		√(p)𝗆^1/3/𝗍^1/6(√(𝗍/p)-√(𝗆))^2/3/(√(p𝗍)-√(𝗆))^1/3
    		;
    𝗏_2=
    		𝗏_2
    		(𝗆,𝗍)
    		   ϱ z_cr
    		=
    		√(p)/(1-p)(√(p𝗍)-√(𝗆))^2/3
    		(√(𝗍/p)-√(𝗆))^2/3/(𝗆𝗍)^1/6

turns the right-hand side of (<ref>) into the Tracy–Widom GUE distribution F_2(α+β).
This completes the proof of <Ref>.





alpha





A. H. Morales, Université du Québec à Montréal, Montreal, QC, Canada and University of Massachusetts Amherst, Amherst, MA, USA

E-mail: 



G. Panova, University of Southern California, Los Angeles, CA, USA

E-mail: 



L. Petrov, University of Virginia, Charlottesville, VA, USA

E-mail: 



D. Yeliussizov, Kazakh–British Technical University, Almaty, Kazakhstan

E-mail: 













 Paper Title: Equivariant $γ$-positivity of Chow rings and augmented Chow rings of matroids.



Leaf Angle Estimation using Mask R-CNN and LETR Vision Transformer


    Venkat Margapuri
Department of Computing Sciences 

Villanova University

Villanova, PA, USA 


Prapti Thapaliya
Department of Computing Sciences 

Villanova University

Villanova, PA, USA 


Trevor Rife
Pee Dee Research and Education Center  

Clemson University

Florence, SC, USA 



    August 5, 2024
==================================================================================================================================================================================================================================================================================================



§ ABSTRACT 
In this paper, we prove the Chow ring and augmented Chow ring of a matroid is equivariant γ-positivity under the action of any group of automorphisms of the matroid. This verifies a conjecture of Angarone, Nathanson, and Reiner. Our method gives an explicit interpretation to the coefficients of the equivariant γ-expansion. Applying our theorem to uniform matroids, we extend and recover several known results regarding uniform matroids and Eulerian quasisymmetric functions. Using these results, we are able to answer a problem proposed by Athanasiadis about extending the γ-expansion of the binomial Eulerian polynomial. 




section[0em]0em0.5pc
subsection[0.5em]0em0.5pc

 




§ NOTATION

We fix some notations that we will use later. For any positive integer n and integers 1≤ a,b≤ n,

    
  * [n]{1,2,…,n} and 2^[n] is the collection of all subsets of [n].
    
  * [a,b]{a,a+1,…,b-1,b} if a≤ b, and [a,b]∅ if a>b.
    
  * [n]_t 1+t+…+t^n-1 for n≥ 1 and [0]_t 0.   

For S⊆ [n], let (S) denote the collection of subsets of S containing no consecutive integers.





§ INTRODUCTION

Given a finite sequence, there are many features that one can ask whether the sequence possesses. A sequence {a_i}_i=0^d or a polynomial f(t)=∑_i=0^d a_i t^i of degree d is said to be palindromic if 

    a_j=a_d-j   for all  j=0,1,…, d.

It is said to be unimodal if there is some 0≤ j≤ d such that 

    a_0≤ a_1≤⋯≤ a_j≥⋯≥ a_d-1≥ a_d.

A polynomial f(t) is palindromic and unimodal if and only if it can be expressed uniquely as 

    f(t)=∑_k=0^⌊d/2⌋γ_kt^k(1+t)^d-2k.

Then we say the polynomial f(t) is γ-positive (or γ-nonnegative) if γ_k≥ 0 for all k. It is not hard to see that a polynomial being γ-positive implies that it is both palindromic and unimodal. Although the γ-positivity is interesting to combinatorialists in its own right, this property has a connection to Gal's conjecture in discrete geometry, and hence has drawn much attention from researchers in combinatorics, discrete geometry, and commutative algebra. For more detail about γ-positivity, we refer the readers to the survey <cit.> by Athanasiadis.

Let M be a loopless matroid on the ground set E. The Chow ring of the matroid M encodes the information of the lattice of flats Ł(M) and is defined as 

    A(M)ℝ[x_F: F∈Ł(M)∖{∅}]/(I+J)

where I=(x_Fx_F':F,F'  not comparable) and J=(∑_F:i∈ x_Fx_F: i∈ E). The augmented Chow ring of M is an extension of A(M) encoding both information from Ł(M) and the independence complex (M), and is defined as

    A(M)ℝ[{x_F:F∈Ł(M)∖{[n]}}∪{y_i:i∈ E}]/(I+J)
   
where I=(x_Fx_F':F,F'  not comparable)+(y_ix_F: i∉ F) and J=(y_i-∑_F:i∉ Fx_F: i∈ E).
Despite the seemingly complicated presentation, the Chow ring A(M) has a very nice ℝ-basis called the Feichtner-Yuzvinsky basis

    FY(M)={x_F_1^a_1x_F_2^a_2… x_F_ℓ^a_ℓ: ∅=F_0⊊ F_1⊊ F_2⊊…⊊ F_ℓ for  0≤ℓ≤ n 
    
        1≤ a_i≤_M(F_i)-_M(F_i-1)-1},

which came from the Gröbner basis of the ideal I+J found by Feichtner and Yuzvinsky <cit.>. There is a similar Feichtner-Yuzvinsky basis for the augmented Chow ring A(M), 

    FY(M){x_F_1^a_1x_F_2^a_2… x_F_ℓ^a_ℓ: ∅⊊ F_1⊊ F_2⊊…⊊ F_ℓ for 0≤ℓ≤ n
    
        1≤ a_1≤_M(F_1),
     
        1≤ a_i≤_M(F_i)-_M(F_i-1)-1  for i≥ 2},

found by the author <cit.> and independently by Eur, Huh, and Larson <cit.>. The Chow rings and augmented Chow rings of matroids play important roles in the settlement of the long-standing Rota-Welsh conjecture and Dowling-Wilson top-heavy conjecture respectively. They are shown to satisfy the so-called Kähler package (see <cit.> for Chow rings and <cit.> for augmented Chow rings). In this package, the Poincaré duality implies that the Hilbert series of the two rings are palindromic, and the hard Lefschetz theorem implies that the Hilbert series are unimodal. The γ-positivity of the Hilbert series was shown by Ferroni, Matherne, Stevens, and Vecchi <cit.>, and independently by Wang <cit.>, using the semismall decomposition introduced in <cit.>. However, no interpretation of the γ_k coefficients is known. In <cit.>, Angarone, Nathanson, and Reiner further conjectured that the matroid Chow rings, under the action of groups of automorphisms of the corresponding matroid, are equivariant γ-positive (see Section <ref> for the definition). In this paper, we give a proof of this conjecture using only the Feichtenr-Yuzvinsky bases and a result in Stanley's classical paper <cit.> (see Theorem <ref>). Our method explicitly interprets the γ-coefficients in terms of the homology group of the rank-selected subposet of the lattice of flats. These are done in Section <ref>. In Section <ref>, we consider the case of uniform matroids. We recover and extend several known results regarding the γ-expansion of the (augmented) Chow rings of uniform matroids and Eulerian quasisymmetric functions in the author's previous work <cit.>, Shareshisan and Wachs <cit.> and Athanasiadis <cit.>. In Section <ref>, we apply results in Section <ref> to answer a problem raised by Athanasiadis <cit.> about finding certain (p,q)-analog of the γ-expansion of the binomial Eulerian polynomial.  

In a recent paper, Stump <cit.> also independently proves the (non-equivariant) γ-positivity for Chow rings and augmented Chow rings of matroids using certain evaluation of the Poincaré extended 𝐚𝐛-index, which Dorpalen-Barry, Maglione, and Stump introduced in <cit.>. In future work, we would like to study the relationship between the Feichtner-Yuzvinsky basis and the Poincaré extended 𝐚𝐛-index, and this extended 𝐚𝐛-index in the equivariant setting. 




§ GROUP ACTIONS ON POSETS

Let P be a finite poset with a unique minimal element 0̂ and a unique maximal element 1̂. If P is graded of rank n with rank function _P: P⟶{0,1,…,n}, then for S⊆ [n-1], the rank selected subposet of P is

    P_S{x∈ P : _P(x)∈ S}∪{0̂,1̂}.

Consider a group G of automorphisms of P acting on the poset P. The action of G preserves the rank of the elements in P; hence for any S⊆ [n-1], the group G permutes the maximal chains in P_S. Let us denote α_P(S) the permutation representation of G generated by the maximal chains in P_S. Consider the virtual representation

    β_P(S)∑_T⊆ S(-1)^|S|-|T|α_P(T).

By Möbius inversion, we also have

    α_P(S)=∑_T⊆ Sβ_P(T).



 
If P is Cohen-Macaulay, then β_P(S) is a genuine representation of G and 

    β_P(S)≅_G H̃_|S|-1(P_S).


Throughout this paper, the poset P we care about is always the lattice of flats of a matroid. It is well known that the lattice of flats of a matroid is always Cohen-Macaulay.





§ EQUIVARIANT Γ-POSITIVITY


Let G be a finite group. For a ℂG-module M, let [M] be the isomorphism class of ℂG-modules containing M. 


The Grothendieck ring R_ℂ(G) of ℂG-modules is a free abelian group having the transversal of isomorphism classes of simple ℂG-modules {[S_i],…,[S_cc(G)]} as a ℤ-basis, where cc(G) is the number of conjugacy classes of G, with the addition and multiplication relations given by 

    [S_i]+[S_j] [S_i⊕ S_j]    and     [S_i]·[S_j][S_i⊗_ℂ S_j]

for 1≤ i,j≤ cc(G) and extended linearly over ℂ. Now every element A in R_ℂ(G) has a unique expression as a=∑_i=1^cc(G)a_i[S_i]. We say A is a genuine representation of G if a_i≥ 0 for all i, denoted by A≥_R_ℂ(G)0.


Given a graded ℂG-module V=⊕_i V_i, define the equivariant Hilbert series of V to be the formal power series 

    _G(V,t)=∑_i[V_i] t^i∈ R_ℂ(G)[[t]].

 

For a finitly dimensional graded ℂG-module V=⊕_i=0^d V_i, we say that V is (G- )equivarant γ-positive if its equivariant Hilbert series can be expressed as 

    _G(V,t)=∑_i=0^d[V_i]t^i=∑_k=0^⌊d/2⌋γ_k t^k(1+t)^d-2k

and the uniquely defined coefficient γ_k∈ R_ℂ(G) is a class of a genuine representation of G over ℂ for all k, i.e. γ_k≥_R_ℂ(G)0 for all k.    


Let M be a loopless matroid of rank r on the ground set E with the lattice of flats Ł(M). Write A(M)_ℂ A(M)⊗_ℝℂ and A(M)_ℂA(M)⊗_ℝℂ. 

From now on, we let G be a group of automorphisms of matroid M.
Consider the action of G acting on the Chow ring A(M)_ℂ=⊕_i=0^r-1A_ℂ^i which gives each graded piece A_ℂ^i a ℂG-module structure. The following conjecture was proposed by Angarone, Nathanson, and Reiner <cit.>.

 
For any (loopless) matroid M of rank r and any group G of automorphisms of M, the matroid Chow ring A(M)_ℂ=⊕_i=0^r-1A^i_ℂ is (G-)equivariant γ-positive. 


In the following, we will prove Conjecture <ref> and also the equivariant γ-positivity of the augmented Chow ring A(M). It is not hard to check that under the action of G, the Feichtenr-Yuzvinsky bases FY(M) and FY(M) are permutation bases for A(M)_ℂ and A(M)_ℂ, respectively.


 
For any loopless matroid M of rank r on the ground set E with the lattice of flats Ł(M), the Chow ring A(M) is equivariant γ-positive with the following γ-expansion

    _G(A(M)_ℂ,t) 
           =∑_S∈([2,r-1])[H̃_|S|-1(Ł(M)_S)] t^|S|(1+t)^r-1-2|S|
       =∑_k=0^r-1/2(∑_S∈([2,r-1])
     |S|=k[H̃_|S|-1(Ł(M)_S)])t^k(1+t)^r-1-2k.




Theorem <ref> generalizes Theorem 4.9, the (-1)-evaluation, in the author's previous work <cit.>.   



Recall that the permutation basis FY(M) consisting of monomials x_F_1^a_1x_F_2^a_2… x_F_ℓ^a_ℓ for any chains (including empty chain)

    ∅≠ F_1⊊ F_2⊊…⊊ F_ℓ⊆E

such that 1≤ a_j≤(F_j)-(F_j-1)-1 for all j.
For a subset S={s_1<s_2<…<s_ℓ}⊆ [r-1], the permutation ℂG-module generated by chains in Ł(M) whose rank set is S is

    α_Ł(M)(S)
           =ℂG{F_1⊊…⊊ F_ℓ: (F_i)=s_i ∀ i}.
       ≅ℂG{F_1⊊…⊊ F_ℓ⊊ E: (F_i)=s_i ∀ i}

Therefore, A(M) as a ℂG-module is the direct sum of α_Ł(M)(S) with some multiplicities for S⊆ [r-1]. Hence we have 

    _G(A(M)_ℂ,t)=∑_S⊆ [r-1]ϕ_S,r(t)[α_Ł(M)(S)]

for some polynomials ϕ_S,r(t). To see what the polynomial ϕ_S,t(t) is, we consider a map f:FY(M)⟶ 2^[r-1] defined by f(x_F_1^a_1x_F_2^a_2… x_F_ℓ^a_ℓx_E^i)={(F_1),(F_2),…,(F_ℓ)} where all flats F_i are distinct and are not E. Since for any S={s_1<⋯<s_ℓ}⊆ [r-1], 

    f^-1(S)={x_F_1^a_1x_F_2^a_2… x_F_ℓ^a_ℓx_E^i : ∅=F_0⊊ F_1⊊⋯⊊ F_ℓ⊊ E  with (F_j)=s_j
    
        1≤ a_j≤ s_j-s_j-1-1   for j=1,2…,ℓ
    
        0≤ i≤ r-s_ℓ-1
        },

the polynomials that count the multiplicities are

    ϕ_S,r(t)
           =(t+t^2+⋯+t^s_1-1)…(t+t^2+⋯+t^s_ℓ-s_ℓ-1-1)(1+t+⋯+t^r-s_ℓ-1)
       =t^|S|[s_1-1]_t[s_2-s_1-1]_t…[s_ℓ-s_ℓ-1-1]_t[r-s_ℓ]_t.

Note that ϕ_S,r(t)=0 if s_j-s_j-1=1 for some 1≤ j≤ℓ, and [r-s_ℓ]_t=1 if r-1∈ S. Therefore, ϕ_S,r(t) takes nonzero value when S∈([2,r-1]). The identity (<ref>) can be rewritten as follows:

    _G(A(M)_ℂ,t)
           =∑_S∈([2,r-1])ϕ_S,r(t)[α_Ł(M)(S)]
       =∑_S∈([2,r-1])ϕ_S,r(t)∑_T:T⊆ S[β_Ł(M)(T)]
       =∑_T∈([2,r-1])(∑_T⊆ S⊆ [r-1]ϕ_S,r(t))[β_Ł(M)(T)].

Surprisingly, it turns out that  

    ∑_T⊆ S⊆ [r-1]ϕ_S,r(t)=t^|T|(1+t)^r-1-2|T|.

This fact and its proof will be present later in Lemma <ref>. Combining with Theorem <ref>, which states that β_Ł(M)(T) is isomorphic to H̃_|T|-1(Ł(M)_T) as ℂG-modules, we obtain

    _G(A(M)_ℂ,t)
           =∑_T∈([2,r-1])[β_Ł(M)(T)]t^|T|(1+t)^r-1-2|T|
       =∑_T∈([2,r-1])[H̃_|T|-1(Ł(M)_T)]t^|T|(1+t)^r-1-2|T|.





 
For n≥ 2 and any subset T∈([2,n-1]), 

    ∑_T⊆ S⊆ [n-1]ϕ_S,n(t)=t^|T|(1+t)^n-1-2|T|,

where ϕ_S,n(t)=0 if S∉([2,n-1]) and for S={s_1<⋯<s_ℓ}∈([2,n-1]),

    ϕ_S,n(t)=
        
        t^|S|[s_1-1]_t[s_2-s_1-1]_t…[s_ℓ-s_ℓ-1-1]_t     if n-1∈ S
    
        t^|S|[s_1-1]_t[s_2-s_1-1]_t…[s_ℓ-s_ℓ-1-1]_t[n-s_ℓ]_t     if n-1∉ S
        .




To prove Lemma <ref>, we will use a combinatorial model motivated by Angarone, Nathanson, and Reiner's combinatorial proof of the permutation representation lift of the Poincaré duality and the Hard Lefschetz theorem in <cit.>.

 
For n≥ 2, consider sequences 𝐰=w_1 w_2… w_n-1 in three symbols ∙, ×, and a blank space , defined as follows:

    
  (W1) If w_i=∙, then w_i-1=× (hence w_1≠∙ and there is no consecutive ∙ in 𝐰).

Say S={s_1<s_2<…<s_ℓ}⊆([2,n-1]) is the set of indices s with w_s=∙.

    
  (W2) If there is s_i<j<s_i+1 such that w_j=×, then w_j=w_j+1=…=w_s_i+1-1=×.
    
  (W3) If there is s_ℓ<j<n such that w_j=×, then w_j=w_j+1=…=w_n-1=×.

Deonte by 𝒲_n the set of such sequences of length n-1. For 𝐰∈𝒲_n, let (𝐰)={i∈ [n-1]: w_i=∙} and (𝐰) be the number of × in 𝐰.



We show that (<ref>) counts in two ways the sum of t^(𝐰) for sequences 𝐰 in 𝒲_n satisfying (𝐰)⊇ T.

First, for fixed S⊇ T, S={s_1<…<s_ℓ}∈([2,n-1]), a sequence 𝐰=w_1… w_n-1∈𝒲_n with (𝐰)=S has w_s_j=∙ for all j. Between w_s_j-1 and w_s_j, we have w_s_j-1=× and we can choose to place from 0 to s_j-s_j-1-1 consecutive ×s on the left of ×=w_s_j-1. Hence the section between w_s_j-1 and w_s_j contributes t[s_j-s_j-1-1]_t. In total, we have

    ∑_𝐰∈𝒲_n
    (𝐰)=St^(𝐰)   =
        
        t^|S|[s_1-1]_t[s_2-s_1-1]_t…[n-s_ℓ-1-2]_t     if n-1∈ S
    
        t^|S|[s_1-1]_t[s_2-s_1-1]_t…[s_ℓ-s_ℓ-1-1]_t[n-s_ℓ]_t     if n-1∉ S
        
       =ϕ_S,n(t).

Therefore, 

    ∑_𝐰∈𝒲_n
    (𝐰)⊇ Tt^(𝐰)=∑_T⊆ S⊆ [n-1]ϕ_S,n(t).

On the other hand, for any sequence 𝐰=w_1… w_n-1∈𝒲_n with (𝐰)⊇ T, since w_i=∙ for i∈ T, by the definition of the sequence we have w_i-1=× for i∈ T. These × contribute the factor t^|T| on the right hand side of (<ref>). In the remaining n-1-2|T| positions, we can choose to place a × or not, which gives the factor (1+t)^n-1-2|T| on the right-hand side of (<ref>). After this process, there is a unique way to add ∙ so that the sequence is in 𝒲_n. This implies that

    ∑_𝐰∈𝒲_n
    (𝐰)⊇ Tt^(𝐰)=t^|T|(1+t)^n-1-2|T|.
    



Here is an example of the procedure in the proof of Lemma <ref>. Let n=12, T={3,9}⊆([2,11]). 

In the first way of the two-way counting, say S={3,5,9}⊇ T, so we place ∙ at 3, 9 and × at 2, 8; and ∙ at 5, × at 4 as follows:



    1     2     3     4     5    6    7    8     9     10    11

         × 
     ∙     ×     ∙               ×      ∙          

 
 3|c|t[2]_t     2|m1 cm|t[1]_t     4|c|t[3]_t     2|c|[3]_t



Then for example, between w_5=∙ and w_9=∙, we can have either w_6=w_7=blank spaces, w_8=×; or w_6=blank space, w_7=w_8=×; or w_6=w_7=w_8=×. These three cases contribute t+t^2+t^3=t[3]_t. Hence we have

    ∑_𝐰∈𝒲_12
    (𝐰)={3,5,9}t^(𝐰)=t[2]_t· t[1]_t· t[3]_t· [3]_t=t^3[2]_t[1]_t[3]_t[3]_t.

Consider the other case that S={3,9,11}⊇ T. We place ∙ at 3, 9 and × at 2, 8; and ∙ at 11 and × at 10 as follows:



    1     2     3     4     5    6    7    8     9     10    11

         × 
     ∙                               ×      ∙     ×     ∙

 
 3|c|t[2]_t     6|c|t[5]_t     2|c|t[1]_t



In this case, since 11=12-1∈ S, between w_9=∙ and w_11=∙ there is no blank space, hence this section contributes the factor t[1]_t. In total, we have

    ∑_𝐰∈𝒲_12
    (𝐰)={3,9,11}t^(𝐰)=t[2]_t· t[5]_t· t[1]_t=t^3[2]_t [5]_t [1]_t.

In the second way of the two-way counting, we count sequences 𝐰 with w_3=w_9=∙ and w_2=w_8=×. We can choose each of the remaining 7 positions to be either a × or a blank space. For each choice there is exactly one way to add ∙ to complete the sequence in 𝒲_12. Hence this implies 

    ∑_𝐰∈𝒲_12
    (𝐰)⊇{3,9}t^(𝐰)=t^|{2,8}|(1+t)^|{1,4,5,6,7,10,11}|=t^2(1+t)^12-1-2· 2.

For example, say we choose w_1=w_4=w_5=w_10=×, then we have 



    1     2     3     4     5    6    7    8     9     10    11

    ×     × 
     ∙     ×     ×               ×      ∙     ×        
.


The only way to complete the above sequence into a sequence in 𝒲_12 by adding ∙ is as follows:



    1     2     3     4     5    6    7    8     9     10    11

    ×     × 
     ∙     ×     ×     ∙          ×      ∙     ×     ∙






A similar proof to the non-augmented case works for the augmented Chow rings. 


 
For any loopless matroid M of rank r on the ground set E, the augmented Chow ring A(M)_ℂ is equivariant γ-positve with    

    _G(A(M)_ℂ,t) 
           =∑_S∈([r-1])[H̃_|S|-1(Ł(M)_S)] t^|S|(1+t)^r-2|S|
       =∑_k=0^r/2(∑_S∈([r-1])
     |S|=k[H̃_|S|-1(Ł(M)_S)])t^k(1+t)^r-2k.





Theorem <ref> generalizes Theorem 4.12, the (-1)-evaluation, in the author's previous work <cit.>.   




Similar to the non-augmented case, the permutation basis FY(M) consists of monomials x_F_1^a_1x_F_2^a_2… x_F_ℓ^a_ℓ for any chains (including empty chain)

    ∅≠ F_1⊊ F_2⊊…⊊ F_ℓ⊆E

such that 1≤ a_1≤(F_1), 1≤ a_j≤(F_j)-(F_j-1)-1 for all j≥ 2. Therefore, A(M) as a ℂG-module is the direct sum of permutation modules α_Ł(M)(S) with some multiplicities for S⊆ [r-1], say 

    _G(A(M)_ℂ,t)=∑_S⊆[r-1]ψ_S,r(t)[α_Ł(M)(S)]

for some polynomial ψ_S,r(t). Consider a map g:FY(M)⟶ 2^[r-1] defined by 

    g(x_F_1^a_1x_F_2^a_2… x_F_ℓ^a_ℓx_E^i)={(F_1),(F_2),…,(F_ℓ)},

where F_i are all distinct and not E. Since for S={s_1<…<s_ℓ}⊆ [r-1], 

    g^-1(S)=
        {x_F_1^a_1x_F_2^a_2… x_F_ℓ^a_ℓx_E^i: ∅=F_0⊊ F_1⊊⋯⊊ F_ℓ⊊ E  with (F_j)=s_j
    
        1≤ a_1≤ s_1, 1≤ a_j≤ s_j-s_j-1-1   for j=2,3…,ℓ
    
        0≤ i≤ r-s_ℓ-1
        }    if S≠∅
    {x_E^j:0≤ j≤ r}    if S=∅.

Hence the multiplicity polynomial for any S⊆ [r-1] is 

    ψ_S,r(t)=
            t^|S|[s_1]_t[s_2-s_1-1]_t…[s_ℓ-s_ℓ-1-1]_t[r-s_ℓ]_t     if S≠∅
    
            [r+1]_t     if  S=∅.

Similar to the non-augmented case, the polynomial ψ_S,r(t) vanishes when S contains consecutive integers, hence (<ref>) can be rewritten as 

    _G(A(M),t)
           =∑_S∈([r-1])ψ_S,r(t)[α_Ł(M)(S)]
       =∑_S∈([r-1])ψ_S,r(t)∑_T⊆ S[β_Ł(M)(T)]
       =∑_T∈([r-1])(∑_T⊆ S⊆ [r-1]ψ_S,r(t))[β_Ł(M)(T)].

It turns out that 

    ∑_T⊆ S⊆ [r-1]ψ_S,r(t)=t^|T|(1+t)^r-2|T|,

of which the proof will be presented after Lemma <ref>. Therefore, we obtain

    _G(A(M)_ℂ,t)
           =∑_T∈([r-1])[β_Ł(M)(T)]t^|T|(1+t)^r-2|T|
       =∑_T∈([r-1])[H̃_|T|-1(Ł(M)_T)]t^|T|(1+t)^r-2|T|,

where the last equality follows from Theorem <ref>.


 
For n≥ 2 and subset T∈([n-1]), 

    ∑_T⊆ S⊆ [n-1]ψ_S,n(t)=t^|T|(1+t)^n-2|T|

where ψ_S,n(t)=0 if S∉([n-1]) and for S={s_1<⋯<s_ℓ}∈([n-1]),

    ψ_S,n(t)=
            t^|S|[s_1]_t[s_2-s_1-1]_t…[s_ℓ-s_ℓ-1-1]_t      if n-1∈ S
    
            t^|S|[s_1]_t[s_2-s_1-1]_t…[s_ℓ-s_ℓ-1-1]_t[n-s_ℓ]_t      if n-1∉ S≠∅
    
            [n+1]_t      if S=∅.



We use another combinatorial model slightly different from 𝒲_n in the non-augmented case to prove Lemma <ref>. 


For n≥ 2, let 𝒲_n be the set of sequences 𝐰=w_0w_1w_2… w_n-1 of length n in three symbols ∙, ×, and a blank space, which also satisfy (W1), (W2), and (W3) in Definition <ref>.

Note that (W1) implies that for any 𝐰∈𝒲_n, w_0≠∙ and there is no consecutive ∙ in 𝐰. We can define (𝐰) and (𝐰) for 𝐰∈𝒲_n as in Definition <ref>. 


The proof is similar to that of Lemma <ref>. We show that (<ref>) is obtained by counting ∑_𝐰∈𝒲_n
(𝐰)⊇ Tt^(𝐰) in two ways.

First, for any fixed S⊇ T, S={s_1<⋯<s_ℓ}∈([n-1]), one can show that 

    ∑_𝐰∈𝒲_n
    (𝐰)=St^(w)   =
        
            t^|S|[s_1]_t[s_2-s_1-1]_t…[s_ℓ-s_ℓ-1-1]_t      if n-1∈ S
    
            t^|S|[s_1]_t[s_2-s_1-1]_t…[s_ℓ-s_ℓ-1-1]_t[n-s_ℓ]_t      if n-1∉ S≠∅
    
            [n+1]_t      if S=∅
       =ψ_S,n(t).

Hence 

    ∑_𝐰∈𝒲_n
    (𝐰)⊇ Tt^(w)=∑_T⊆ S⊆ [n-1]ψ_S,n(t).

On the other hand, to count t^(𝐰) over 𝐰=w_0… w_n∈𝒲_n with (𝐰)⊇ T, fix w_i=∙ and w_i-1=× for all i∈ T first. This gives the factor t^|T| in the right-hand side of (<ref>). Then in the remaining n-2|T| positions, we can choose to place a × or not. Any such selection of the remaining n-2|T| spots, there is a unique way to add × so that the resulting sequence is in 𝒲_n. These selections contribute the factor (1+t)^n-2|T| in the right-hand-side of (<ref>). Therefore, we have

    ∑_𝐰∈𝒲_n
    (𝐰)⊇ Tt^(𝐰)=t^|T|(1+t)^n-2|T|.





§ UNIFORM MATROIDS AND Q-UNIFORM MATROIDS
 
In this section, we consider the special case that the matroid M is a uniform matroid U_r,n of rank r and G is the symmetric group _n. In this case, the lattice of flats is some rank-selected subposet of the Boolean lattice B_n=(2^[n],⊆). The equivariant γ-coefficient H̃_|R|-1((B_n)_R) has a well-studied ℂ_n-module structure for any R⊆[n-1]. For the terminology of representation of _n, standard Young tableaux, and symmetric functions in this section, we refer the readers to Wachs <cit.> and Stanley <cit.>. For the introduction to the permutation statistics such as , , , etc., see Chapter 1 of Stanley <cit.>.

Let  be the Frobenius characteristic map that sends an isomorphism class of ℂ_n-module to a symmetric function of degree n. Let  be the stable principle specialization that replaces x_i by q^i-1 in a symmetric function. When considering the representation of _n, instead of the equivariant Hilbert series, we can also use the graded Frobenius series to encode a graded representation of _n as a polynomial with symmetric function coefficients. For a graded ℂ_n-module V=⊕_i V_i, the graded Frobenius series of V is defined to be (V,t)∑_i(V_i)t^i.

Let λ, μ be two partitions such that μ⊆λ (i.e. μ_i≤λ_i for all i). The skew shape λ/μ is said to be a ribbon (or a skew hook or a border strip) if its corresponding Ferrer diagram does not contain any 2× 2 subdiagram. A ribbon H_R,n can be described in terms of the number n of its cells and its descent set R⊆ [n-1]. Given a ribbon of n cells, we label the its cells with numbers 1 through n, starting from the southwestern end cell, through the adjacent cell, ending at the northeastern end cell. We say a cell i is a descent of the ribbon if cell i+1 is directly above cell i. Then the collection of all its descents is called its descent set.


Let n=7. Then the diagrams of H_{2,4},7 and H_{1,3},7 are 

    [                                                   H_{2,4},7                                                                                                                                                                                                                                       H_{1,3},7;                                                                                                                                                                                                                                                                                                            ; 5     6     7
        3     *(green)4
     
        1         *(green)2                                                                                                                                                                                         4     5     6     7
    
        2         *(green)3 
    (green) 1 ].


Let S^λ/μ be the Specht module of skew shape λ/μ. If λ/μ is a ribbon H_R,n, the corresponding Specht module S^H_R,n is known as a Foulkes representation. We summarize some facts of this representation from  <cit.> and <cit.> in the following theorem. 

For a partition λ of n, denoted by λ⊢ n, let SYT(λ) be the set of standard Young tableaux of shape λ. For P∈ SYT(λ), let (P) be the set of entries (called descents) i∈[n-1] such that i+1 appears in a lower row than i in P, and (P)=|(P)|.

 
Let R⊆ [n-1]. 

    
  (i) Then

    H̃_|R|-1((B_n)_R)≅__n S^H_R,n≅__n⊕_λ⊢ nb_λ(R)S^λ,
    
where b_λ(R) is the number of P∈ SYT(λ) for which (P)=R.
    
  (ii) Let ℙ denote the set of positive integers and ℙ_n the set of words of length n over ℙ. For w=w_1… w_n∈ℙ_n, we say i∈(w) if w_i>w_i+1 and write _w=x_w_1… x_w_n. Then the Frobenius characteristic is the ribbon Schur function

    (H̃_|R|-1((B_n)_R))=s_H_R,n=∑_w∈ℙ_n
    (w)=R𝐱_w,










Applying Theorem <ref> to the uniform matroids U_r,n together with Theorem <ref>, one obtains the graded Frobenius series of A(U_r,n)_ℂ. Applying the operator  ∏_i=1^n(1-q^i)(-) to the graded Frobenius series gives the Hilbert series of q-Uniform matroid U_r,n(q) (See Section 3 in <cit.> for the explanation). Note that the stable principle specialization on the ribbon Schur function gives

    ∏_i=1^n(1-q^i)(s_R,n)=∑_σ∈_n 
    (σ)=Rq^(σ^-1)=∑_σ∈_n 
    (σ)=Rq^(σ)

(see <cit.> and its proof for detailed discussion of the above identity). We obtain the following corollary.

 
For any positive integer n and 1≤ r≤ n,

    (A(U_r,n)_ℂ,t)
           =∑_R∈([2,r-1])s_H_R,nt^|R|(1+t)^n-1-2|R|
       =∑_k=0^⌊r-1/2⌋(∑_R∈([2,r-1])
     |R|=ks_H_R,n)t^k(1+t)^r-1-2k

and 

    (A(U_r,n(q)),t)
        =∑_k=0^⌊r-1/2⌋ξ_r,n,k(q)t^k(1+t)^r-1-2k

where 

    ξ_r,n,k(q)=∑_σq^(σ^-1)=∑_σq^(σ)

and the sum runs through all σ∈_n for which (σ)∈([2,r-1]) has k elements.



Corollary <ref> generalizes Corollary 4.10 and 4.11 in the author's previous work <cit.>    


Applying Theorem <ref> and the similar procedures to U_r,n gives the following:

 
For any positive integer n and 1≤ r≤ n,

    (A(U_r,n)_ℂ,t)
           =∑_R∈([r-1])s_H_R,nt^|R|(1+t)^n-1-2|R|
       =∑_k=0^⌊r/2⌋(∑_R∈([r-1])
     |R|=ks_H_R,n)t^k(1+t)^r-2k

and 

    (A(U_r,n)_ℂ,t)=∑_k=0^⌊r/2⌋ξ_r,n,k(q)t^k(1+t)^r-2k,

where 

    ξ_r,n,k(q)=∑_σq^(σ^-1)=∑_σq^(σ)

and the sum runs through all σ∈_n for which (σ)∈([r-1]) has k elements.



Corollary <ref> generalizes Corollary 4.13 and 4.14 in the author's previous work in <cit.>.


Recall from Corollary 4.2 and 4.4 in <cit.> that

    (A(U_n-1,n),t)=t^-1Q_n^0(,t),     
       (A(U_n,n),t)=(A(U_n-1,n),t)=Q_n(,t), 
       (A(U_n,n),t)=Q_n(,t).

Here Q_n^0, Q_n, Q_n are Eulerian quasisymmetric functions introduced by Shareshian and Wachs <cit.>, which can be defined as

    Q_n^0(,t)=∑_σ∈ D_nF_(σ),n()t^(σ),    Q_n(,t)=∑_σ∈_nF_(σ),n()t^(σ),

and

    Q_n(,t)=∑_σ∈_nF_(σ),n()t^(σ)+1,

where F_S,n() for S⊆[n-1] is Gessel's fundamental quasisymmetric function [Note that our definition of the fundamental quasisymmetric function is the reverse of the usual one. The usual definition is summed over i_1≤ i_2≤⋯≤ i_n with j∈ S implying i_j<i_j+1.]

    F_S,n()∑_i_1≥ i_2≥⋯≥ i_n
     j∈ S⇒ i_j>i_j+1x_i_1x_i_2… x_i_n   with F_ϕ,0 1,

and D_n is the subset of derangements of _n, and _n is the set of decorated permutations satisfying D_n⊂_n⊂_n. 
See <cit.> for the definition of the set statistic  over _n and more detail of these symmetric functions. From Lemma 6.20 in <cit.>, applying ∏_i=1^n(1-q^i)(-) to Q_n^0, Q_n, and Q_n yields the q-analogs of the Eulerian type polynomials

    d_n(q,t)=∑_σ∈ D_nq^(σ)-(σ)t^(σ),    A_n(q,t)=∑_σ∈_nq^(σ)-(σ)t^(σ),

and 

    A_n(q,t) 1+∑_k=1^nnkA_k(q,t) =∑_σ∈_nq^(σ)-(σ)t^(σ)+1.



Let NDD_n be the set of words w∈ℙ_n with no double descents, i.e. (w)∈([n-1]). Combining (<ref>) and Corollary <ref>, we have:
 
For positive integer n≥ 2, we have 

    Q_n^0(,t)
           =∑_R∈([2,n-2])s_H_R,nt^|R|(1+t)^n-1-2|R|
       =∑_w∈ NDD_n
     w_1≤ w_2, w_n-1≤ w_n_w t^(w)+1(1+t)^n-2-2(w)
       =∑_k=0^⌊n-2/2⌋ξ_n,k^0() t^k+1(1+t)^n-2-2k,

where 

    ξ_n,k^0()=∑_R∈([2,n-2])
     |R|=ks_H_R,n=∑_w∈ NDD_n
     w_1≤ w_2, w_n-1≤ w_n 
    (w)=k_w.

And

    d_n(q,t)=∑_k=0^⌊n-2/2⌋ξ_n,k^0(q)t^k+1(1+t)^n-2-2k,

where

    ξ_n,k^0(q)=∑_σq^(σ^-1)=∑_σq^(σ)

and both of the sums run through σ∈ D_n which has k descents and no double descents, no first, no last descent.


Combining (<ref>) with Corollary <ref> and Corollary <ref> yields the following:

 
For positive integer n≥ 1, we have the following expressions

    Q_n(,t)
           =∑_R∈([2,n-1])s_H_R,nt^|R|(1+t)^n-1-2|R|=∑_R∈([n-2])s_H_R,nt^|R|(1+t)^n-1-2|R|
       =∑_w∈ NDD_n
     w_1≤ w_2_w t^(w)(1+t)^n-1-2(w)=∑_w∈ NDD_n
     w_n-1≤ w_n_w t^(w)(1+t)^n-1-2(w)
       =∑_k=0^⌊n-1/2⌋ξ_n,k() t^k(1+t)^n-1-2k,

where 

    ξ_n,k()
           =∑_R∈([2,n-1])
     |R|=ks_H_R,n
        =∑_R∈([n-2])
     |R|=ks_H_R,n
       =∑_w∈ NDD_n
     w_1≤ w_2
    (w)=k_w
        =∑_w∈ NDD_n
     w_n-1≤ w_n
    (w)=k_w.

And 

    A_n(q,t)=∑_k=0^⌊n-1/2⌋ξ_n,k(q)t^k+1(1+t)^n-1-2k,

where 

    ξ_n,k(q)=∑_σq^(σ^-1)=∑_σq^(σ)

and both of the sums run through either:

    
  * σ∈_n with k descents and no double descents, no first descent.
    
  * σ∈_n with k descents and no double descents, no last descent.




In <cit.>, the author summarizes different interpretations of the γ-coefficients for the Eulerian polynomial. In particular, the interpretations are given by counting permutations in _n related to descent set in ([2,n-1]) or in ([n-2]). This fact is lifted to the q-analog and the symmetric function analog in Corollary <ref>. The agreement of the two interpretations regarding ([2,n-1]) and ([n-2]) was somehow shown algebraically by (<ref>).   


Combining (<ref>) and Corollary <ref> gives the following:

 
For positive integer n≥ 1, we have 

    Q_n(,t)
           =∑_R∈([n-1])s_H_R,nt^|R|(1+t)^n-2|R|
       =∑_w∈ NDD_n_w t^(w)(1+t)^n-2(w)
       =∑_k=0^⌊n/2⌋ξ_n,k() t^k(1+t)^n-2k,
 
where 

    ξ_n,k()=∑_R∈([n-1])
     |R|=ks_H_R,n=∑_w∈ NDD_n 
    (w)=k_w.

And 

    A_n(q,t)=∑_k=0^⌊n/2⌋ξ_n,k(q)t^k(1+t)^n-2k,

where

    ξ_n,k(q)=∑_σq^(σ^-1)=∑_σq^(σ)

and both of the sums run through σ∈_n with k descents and no double descents.




§ A PROBLEM FROM ATHANASIADIS
 
In this section, we apply some results from Section <ref> to solve a problem proposed by Athanasiadis <cit.>.

For σ∈_n, we set  

    ^*(σ)=(σ)     if σ_1=1
    (σ)-1      if σ_1>1
        .

Recall the following (p,q)-analog of the γ-expansion of the Eulerian polynomial A_n(t) in <cit.>.
 
For all positive integer n≥ 1, 

    ∑_σ∈_n p^^*(σ)q^(σ)-(σ)t^(σ)=∑_k=0^⌊n-1/2⌋ξ_n,k(p,q)t^k(1+t)^n-1-2k
    
where 

    ξ_n,k(p,q)=∑_σp^(σ^-1)q^(σ^-1)

and the sum runs through σ∈_n with k descents and no double descents, no last descent.


Recall a similar q-analog of the γ-expansion of the binomial Eulerian polynomial A_n(t) found by Shareshian and Wachs <cit.> is given in Corollary <ref>. Athanasiadis proposed the following problem.
[<cit.>]
Find a p-analog of the q-polynomial in Corollary <ref> that is similar to Theorem <ref>.   


The idea of finding such a (p,q)-analog of A_n(t) is similar to the proof of Theorem <ref>. But the key step is using (<ref>) [This interpretation is proved in the author's previous work <cit.>. The original definition given by Shareshian and Wachs <cit.> is 
Q_n(,t) h_n+t∑_k=1^n h_n-k Q_k(,t).], which interprets the expansion of Q_n(,t) into fundamental quasisymmetric functions in terms of decorated permutations _n.


Write Q_n(,t)=∑_λ⊢ n N_λ(t)s_λ(). The following theorem is analogous to Corollary 2.41 in <cit.>.

 
For λ⊢ n, 

    N_λ(t)=∑_P∈ SYT(λ)
    (P)∈([n-1]) t^(P)(1+t)^n-2(P)=∑_k=0^⌊n/2⌋ξ_λ,k  t^k(1+t)^n-2k

where ξ_λ,k is the number of tableaux P∈ SYT(λ) for which (P)∈([n-1]) has k elements.



Apply Corollary <ref>. Then use Theorem <ref> (i) to expand s_H_R,n into Schur functions of regular shapes. We have

    Q_n(,t)=∑_λ⊢ n(∑_R∈([n-1])b_λ(R)t^|R|(1+t)^n-2|R|)s_λ().

Comparing the coefficient of s_λ(), we obtain

    N_λ(t)=∑_R∈([n-1])b_λ(R)t^|R|(1+t)^n-2|R|
        =∑_P∈ SYT(λ)
    (P)∈([n-1])t^(P)(1+t)^n-2(P)



We used stable principle specialization to obtain many interesting q-analogs in the previous sections. To obtain a (p,q)-analog of A_n(t), we need a different specialization. Given a formal power series f()=f(x_1,x_2,…) and a positive integer m≥ 1, the principal specialization (f) of f is obtain from f by setting

    x_i=
            q^i-1    if 1≤ i≤ m
    
            0           if i≥ m+1
        .

Then the principal specialization of F_S,n (see <cit.>) and s_λ() (see <cit.>) are given by the formulas

    ∑_m≥ 1(F_S,n())p^m-1=p^|S|q^∑_i∈ Si/(1-p)(1-pq)…(1-pq^n)

and

    ∑_m≥ 1(s_λ())p^m-1=∑_P∈ SYT(λ)p^(P)q^(P)/(1-p)(1-pq)…(1-pq^n).


For any decorated permutation σ∈_n, we set 

    (σ)=(σ)     if σ_1=0  or 1
    (σ)-1     if σ_1>1
        .


Now we are ready to states our (p,q)-analog for A_n(t).

For all positive integer n≥ 1,

    ∑_σ∈_np^(σ)q^(σ)-(σ)t^(σ)+1   =∑_σ∈_n 
    (σ)∈([n-1])p^(σ^-1)q^(σ^-1)t^(σ)(1+t)^n-2(σ)
       =∑_k=0^⌊n/2⌋ξ_n,k(p,q) t^k(1+t)^n-2k

where 

    ξ_n,k(p,q)=∑_σp^(σ^-1)q^(σ^-1)

and the sum runs through σ∈_n with k descents and no double descents.



 First applying principal specialization on (<ref>) and by (<ref>), we have

    ∑_m≥ 1(Q_n(,t))p^m-1
    
        =   ∑_σ∈_n∑_m≥ 1(F_(σ),n())p^m-1t^(σ)+1
    
        =   ∑_σ∈_np^(σ)q^(σ)-(σ)t^(σ)+1/(1-p)(1-pq)…(1-pq^n)  (by Lemma 6.20 in <cit.>).

On the other hand, 

    ∑_m≥ 1(Q_n(,t))p^m-1
    
        =   ∑_λ⊢ n N_λ(t)∑_m≥ 1(s_λ())p^m-1
    
        =   ∑_λ⊢ n N_λ(t)∑_P∈ SYT(λ)p^(P)q^(P)/(1-p)(1-pq)…(1-pq^n)  (by (<ref>)).

Therefore, we have 

    ∑_σ∈_np^(σ)q^(σ)-(σ)t^(σ)+1
        =∑_λ⊢ n∑_P∈ SYT(λ) N_λ(t)p^(P)q^(P)
    
        =   ∑_λ⊢ n∑_P∈ SYT(λ)∑_Q∈ SYT(λ)
    (Q)∈([n-1])p^(P)q^(P) t^(Q)(1+t)^n-2(Q)  (by Theorem <ref>)
    
        =   ∑_σ∈_n
    (σ)∈([n-1])p^(σ^-1)q^(σ^-1)t^(σ)(1+t)^n-2(σ).

The last equality is obtained by applying the RSK algorithm in Section 7.11 and Lemma 7.23.1 in Stanley <cit.>.




§ FINAL REMARK


Besides the uniform matroids, another class of matroids that our γ-coefficient interpretation seems to give some interesting results is the Braid matroids K_n. It is the matroid associated with the hyperplane arrangement whose intersection lattice is given by the partition lattice Π_n, or equivalently, the graphical matroid of the complete graph. To the best of our knowledge, there are no general formulas known for the Hilbert series of Chow ring A(K_n) and augmented Chow ring A(K_n). Applying Theorem <ref> and Theorem <ref>, we have

    _G(A(K_n),t)=∑_S⊆([2,n-2])[H̃_|S|-1((Π_n)_S)] t^|S|(1+t)^n-2-2|S|

and 

    _G(A(K_n),t)=∑_S⊆([n-2])[H̃_|S|-1((Π_n)_S)] t^|S|(1+t)^n-1-2|S|.

These give an explicit connection between the (augmented) Chow ring of the braid matroid K_n and the top reduced homology of the rank-selected partition lattice (Π_n)_S. Though the characterization of the structure of H̃_|S|-1((Π_n)_S) is also an open problem proposed by Stanley <cit.>, and still remains open for most of S⊆ [n-2] nowadays, any development on either of the problems might shed light on the other one.


§ ACKNOWLEDGEMENTS

The author thanks Victor Reiner for suggesting the problem of extending the equivariant Charney-Davis quantity results in <cit.> which leads to the genesis of this paper. 



abbrv





 Paper Title: On the Uncrossed Number of Graphs.



NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elements
    [
    
=========================================================================================================




§ ABSTRACT 

Visualizing a graph G in the plane nicely, for example, without crossings, is unfortunately not always possible.
To address this problem, Masařík and Hliněný <cit.> recently asked for each edge of G to be drawn without crossings while allowing multiple different drawings of G.
More formally, a collection 𝒟 of drawings of G is uncrossed if, for each edge e of G, there is a drawing in 𝒟 such that e is uncrossed.
The uncrossed number (G) of G is then the minimum number of drawings in some uncrossed collection of G.

No exact values of the uncrossed numbers have been determined yet, not even for simple graph classes.
In this paper, we provide the exact values for uncrossed numbers of complete and complete bipartite graphs, partly confirming and partly refuting a conjecture posed by Hliněný and Masařík <cit.>.
We also present a strong general lower bound on (G) in terms of the number of vertices and edges of G.
Moreover, we prove -hardness of the related problem of determining the edge crossing number of a graph G, which is the smallest number of edges of G taken over all drawings of G that participate in a crossing.
This problem was posed as open by Schaefer in his book <cit.>.



  






§ INTRODUCTION


In a drawing of a graph G, the vertices are represented by distinct points in the plane and each edge corresponds to a simple continuous arc connecting the images of its end-vertices.
As usual, we identify the vertices and their images, as well as the edges and the line segments representing them.
We require that the edges
pass through no vertices other than their endpoints. We assume for simplicity that any
two edges have only finitely many points in common, no two edges touch at an interior point
and no three edges meet at a common interior point.

A crossing in a drawing D of G is a common interior point of two edges of D where they properly cross.
For a drawing D of a graph G, we say that an edge e of D is uncrossed in D if it does not share a crossing with any other edge of D.

There are two staple problems in the graph drawing field that defined the past eighty years of development in the area.
The first one, dating back to World War II times <cit.>, is the problem of determining the crossing number (G) of a graph G, defined as the smallest number of crossings required in any drawing of G in the plane.
The crossing number problem has been intensively studied ever since, especially in the past thirty years.
Computing the crossing number is -hard on general graphs <cit.>, and one can find a thorough overview of the area in a recent book by Schaefer <cit.>.

The second, only slightly newer problem, is that of determining the thickness θ(G) of a graph G, defined as the smallest integer k such that G can be edge-partitioned into k planar graphs.
This problem was proposed by Harary <cit.> in 1961 and since then this concept has played an important role in graph drawing. 
Unlike for planarity, deciding whether a graph is biplanar, that is whether θ(G)≤ 2, is -complete <cit.>.
For an overview of the progress up to 1998, consult a survey by Mutzel, Odenthal, and Scharbrodt <cit.>.


In this paper, we investigate a very recent notion inspired by a fusion of both concepts into one.
We say that a collection 𝒟(G) of drawings of G is uncrossed if for each edge e of G there is at least one drawing in 𝒟(G) in which e is uncrossed; see <Ref> for an example.
Hliněný and Masařík <cit.>, in relation to extensions of the traditional crossing number of a graph, defined the uncrossed number (G) of a graph G as the smallest size of an uncrossed collection of drawings of G.



The motivation for the uncrossed number <cit.> is that finding a handful of different drawings of a graph G instead of just one “flawless” drawing shall highlight different aspects of G and thus could be useful for the visualization of G, besides the theoretical interest.
The requirement that each edge is uncrossed in at least one drawing is a natural way to enforce that the drawings will highlight each aspect of the graph as a whole.

Let us also formulate the decision version of the problem of determining (G) of a given graph G.
UncrossedNumberA graph G and a positive integer k.Are there drawings D_1,…,D_k of G such that, for each edge e∈ E(G), there exists an i∈[k] such that e is uncrossed in the drawing D_i?

Clearly, for every graph G, we have

    (G) ≥θ(G),

because the uncrossed edges in each drawing of an uncrossed collection of G induce an edge-partition of G into planar graphs.
However, this new concept significantly differs from thickness (which just partitions the edges of G) in the sense that all edges of G have to be present along with the uncrossed subdrawing in each drawing of our uncrossed collection.
In fact, the requirements of an uncrossed collection bring us close to the related notion of the outerthickness θ_o(G) of G, which is the minimum number of outerplanar graphs into which we can decompose G.

OuterthicknessA graph G and a positive integer k.Can G be decomposed into k outerplanar graphs?

As noted by Hliněný and Masařík <cit.>, given a decomposition {G_1,…,G_k} of G into outerplanar graphs, we can let D_i be an outerplanar drawing of G_i with all remaining edges of G being drawn in the outer face.
This gives us 

    (G) ≤θ_o(G)

for every graph G.
Combining this with a result of Gonçalves <cit.>, which implies θ_o(G)≤2θ(G), we actually obtain the following chain of inequalities

    1/2θ_o(G) ≤θ(G) ≤(G) ≤θ_o(G) ≤ 2 θ(G).



So far, the exact values of uncrossed numbers are not very well understood.
Masařík and Hliněný <cit.> exactly determined (G) of only a few sporadic examples of graphs G, such as (K_7)=3.




 §.§ Our Results


We determine the exact values of uncrossed numbers for specific and natural graph classes.
First, we derive the formula for the uncrossed number of complete graphs.


For every positive integer n, it holds that 

    (K_n) = 
    ⌈n+1/4⌉,    for  n ∉{4,7}
    
        3,    for  n=7
    
        1,    for  n=4.



We also find the exact formula for the uncrossed number of complete bipartite graphs.



For all positive integers m and n with m ≤ n, it holds that 

    (K_m,n) = 
      ⌈mn/2m+n-2⌉,    for  m ≤ n ≤ 2m-2 
    ⌈mn/2m+n-1⌉,    for  n=2m-1 
    ⌈mn/2m+n⌉,    for  6 ≤ 2m ≤ n 
    
        1,    for  m ≤ 2.



Let us mention that the exact values of the thickness θ(K_m,n) of complete bipartite graphs are not known for all values of m and n;  see <cit.> for further discussion.

We compare our formulas on (K_n) and (K_m,n) with known formulas on the outerthickness of K_n and K_m,n.
Hliněný and Masařík <cit.> conjectured the uncrossed numbers and outerthickness to be the same for both complete and complete bipartite graphs except in the planar but not outerplanar cases.
Guy and Nowakowski <cit.> showed that

    θ_o(K_n) = 
    ⌈n+1/4⌉,    for  n ≠ 7 
    
        3,    for  n=7

and 

    θ_o(K_m,n) = ⌈mn/2m+n-2⌉

for all positive integers m and n with m ≤ n. 
Note that it follows from <Ref> that (K_n)=θ_o(K_n) for every n≠ 4.
For n=4, we have (K_4)=1 while θ_o(K_4)=2.
This confirms the conjecture of Hliněný and Masařík <cit.> in the case of complete graphs.

Since 

    ⌈mn/2m+n⌉≤⌈mn/2m+n-2⌉ = ⌈mn/2m+n + 2mn/(2m+n-2)(2m+n)⌉≤⌈mn/2m+n⌉ + 1

for n ≥ 2m-1>1, it follows from <Ref> and <Ref> that the uncrossed number (K_m,n) differs from the outerthickness θ_o(K_m,n) of K_m,n by at most 1.
In particular, we have (K_n,n)=θ_o(K_n,n) for every positive integer n.
However, Theorem <ref> and (<ref>) give, for example, (K_4,7)= 2 and θ_o(K_4,7)=3.
Since K_4,7 is not planar, this refutes the conjecture of Hliněný and Masařík <cit.> in the case of complete bipartite graphs.



Second, we turn our attention to general graphs and their uncrossed number.
We improve the trivial lower bound of (G) ≥⌈ m/(3n-6)⌉ for any graph G with n vertices and m edges.
By carefully balancing between the numbers of edges in uncrossed subdrawings of G and the numbers of edges that can be drawn within faces of uncrossed subdrawings, we derive the following estimate.



Every connected graph G with n ≥ 3 vertices and m ≥ 0 edges satisfies

    (G) ≥⌈m/f(n,m)⌉

where f(n,m) = (3n-5+√((3n-5)^2-4m))/2.


The bound from Theorem <ref> becomes interesting for m ≥ 3n-6.
This is because we then have f(n,m) ≤ 3n-6 for all integers n ≥ 2 as 

    √((3n-5)^2-4m)≤√(9n^2-42n+49) = 3n-7
 for any m ≥ 3n-6 ≥ 0.
It follows that the lower bound from <Ref> is at least as good as (G) ≥⌈ m/(3n-6)⌉ for any connected G with n ≥ 2 vertices and m ≥ 3n-6 edges.

The lower bound from <Ref> gets stronger as the graph G gets denser.
For example, if G contains n vertices and ε n^2 edges for n sufficiently large and ε∈ (0,1/2), we get

    f(n,m) = (3n-5+√((9-4ε)n^2 -30n + 25))/2 ≤ (3+√(9-4ε)) n/2.

Since (3+√(9-4ε))/2 < 3 for ε > 0, we obtain (G) ≥⌈m/c_ε n⌉ for any ε > 0 and some constant c_ε<3, instead of trivial (G) ≥⌈m/3n-6⌉.
We note that the best constant c_ε obtainable from Theorem <ref> is (3+√(7))/2 ∼ 2.82 as ε≤ 1/2.




We also consider computational complexity aspects related to the UncrossedNumber problem. 
As we will see later, a closely related problem is the one of determining the edge crossing number of a given graph G, which is the smallest number of edges involved in crossings taken over all drawings of G.
The notion of the edge crossing number is based on results by Ringel <cit.>, Harborth and Mengersen <cit.>, and Harborth and Thürmann <cit.>.

A graph G and a positive integer k.Is there a drawing D of G with at most k edges involved in crossings?

The complementary problem to  is the following one.

A graph G and a positive integer k.Is there a drawing D of G with at least k edges not involved in any crossings?

In his monograph on crossing numbers, Schaefer <cit.> mentions that the problem of determining the computational complexity of  is open.
Here, we resolve this open question by showing that the problem is -complete.



	
	The  problem is -complete.


By the complementarity of the problems  and , we obtain the following result.


	The  problem is -complete. 


As a consequence of our reduction, we also obtain the following relative result. 


	If the Outerthickness problem is -hard, then also the  problem is -hard.

However, in contrast to the complexity of the thickness problem, which was shown to be -hard already in 1983 by Mansfield <cit.>, the complexity of the Outerthickness problem remains open.









§ PRELIMINARIES




We may, without loss of generality, restrict to only simple graphs in the whole paper.
This is since, in each of the formulated problems, whenever an edge e is a part of an uncrossed subdrawing (as discussed next),
any other edge parallel to e can be drawn uncrossed closely along e, too.

Let D' be a subdrawing of D consisting of only uncrossed edges of D. Note that we do not require D' to contain all such edges.
In this situation, we call D' an uncrossed subdrawing of G and we say that it represents the subgraph of G formed by edges that are drawn in D'.  Precisely, D' is an uncrossed subdrawing of G if there exists a drawing D of a graph G such that D' is formed by a subset of the uncrossed edges of D.




Let D' be an uncrossed subdrawing of a connected graph G.
Then D' is a planar drawing and, for every edge {u,v} of G, the vertices u and v are incident to a common face of D'.
Moreover, there is an uncrossed subdrawing D” of G such that D” represents a connected supergraph of the graph represented by D'.


The drawing D' is clearly planar, as, by the definition of D', each edge of D' is uncrossed in a drawing D of G and thus also in D'.
Moreover, it is a folklore fact that two vertices u and v in a planar drawing, here in D', are not incident to a common face if and only if there exists a cycle C⊆ D' such that u and v are drawn on different sides of C.
In the latter case, however, the edge {u,v} would cross some edge of C in D, which is impossible since no edge of D' is crossed.

We prove the second part by induction on the number of connected components represented by D'.
The case of one component is trivial, as D”=D'.
Otherwise, since G is connected, there exists an edge e={u,v} of G that is not drawn in D' and such that u and v belong to different components represented by D'.
By the first part of the lemma, the vertices u and v are incident to the same face of D'.
So, let D^+ be the planar drawing obtained from D' by adding a crossing-free arc representing the edge e.
Clearly, every edge of G is still incident to a common face of D^+, and so D^+ can be completed into a drawing of G such that D^+ stays uncrossed.
The subgraph of G represented by D^+ has fewer components than we started with, and so we find the desired D” by induction.




For a graph G, let h(G) be the maximum number of uncrossed edges in some drawing D of G. 
Let DW_n be a planar drawing of the wheel graph W_n on n vertices; see <Ref>.
Note that DW_n is unique up to homeomorphism of the sphere and reflection as W_n is 3-connected.
The following result by Ringel <cit.> gives a formula for h(K_n) for every integer n ≥ 4,
and additionally claims that drawings of K_n with the maximum number of uncrossed edges have a unique structure. 



For every integer n ≥ 4, we have
h(K_n) = 2n-2.
Moreover, if D is a drawing of K_n with 2n-2 uncrossed edges, then D contains the drawing DW_n with all edges from D ∖ DW_n being drawn in the outer face of DW_n. 


We also mention an analogous result for the complete bipartite graphs K_m,n, derived by Mengersen <cit.>.



For all positive integers m and n with m ≤ n, we have

    h(K_m,n) = 
    
        2m+n-2,    for  m=n 
    
        2m+n-1,    for  m < n < 2m 
    
        2m+n,    for  2m ≤ n.



The parameter h(G) can be used to estimate the uncrossed number of G.
Let {D_1,…,D_k} be an uncrossed collection of drawings of a graph G that has m edges.
Since every drawing D_i contains at most h(G) edges that are uncrossed by any other edge in D_i, we immediately obtain the following lower bound

    (G) ≥⌈m/h(G)⌉.

This bound together with <Ref> give us quite close estimates for (K_n) and (K_m,n), respectively.
In particular, for n≥ 2 we have

    (K_n) ≥⌈n2/2n-2⌉.

On the other hand, we recall the upper bound (<ref>) on the uncrossed number of an arbitrary graph G using the notion of outerthickness of G.











§ PROOF OF THEOREM 1


In this section, we prove <Ref> by providing the exact formula for the uncrossed number of complete graphs.
That is, we show

    (K_n) = 
    ⌈n+1/4⌉,    for  n ∉{4,7}
    
        3,    for  n=7
    
        1,    for  n=4

for every positive integer n.

We start with the upper bound, which is easier to prove.
For n ∉{4,7}, the upper bound follows from (<ref>) and (<ref>) as we have 

    (K_n) ≤θ_o(K_n) = ⌈n+1/4⌉.

For n=4, we obviously have (K_4)=1 as K_4 is planar.
Finally, (K_7)=3 was proved by Hliněný and Masařík <cit.>.

It remains to prove the lower bound.
Since we already know that (K_7)=3 and (K_4)=1 and the statement is trivial for n ≤ 3, it suffices to consider the case n ≥ 5 with n ≠ 7.
Let {D_1,…,D_k} be an uncrossed collection of drawings of K_n and let D'_1,…,D'_k be corresponding uncrossed subdrawings of K_n such that D'_1∪…∪ D'_k covers E(K_n). 
By (<ref>), 

    (K_n) ≥⌈n2/2n-2⌉.

By <Ref>, we get that if any uncrossed subdrawing D'_i contains 2n-2 edges, then D'_i (as a wheel) contains a universal vertex, that is, a vertex that is adjacent to all remaining vertices in D'_i.
If every drawing D'_i contains at most 2n-3 edges, then 

    (K_n) ≥⌈n2/2n-3⌉ = ⌈n/4 + n/4(2n-3)⌉ = ⌈n+1/4⌉

and we are done.

Thus, suppose that some drawing D'_i contains 2n-2 edges.
Without loss of generality, we can assume i=1.
We then know that D'_1 contains a universal vertex v.
In every drawing D'_j with j > 1, the edges incident to v are already counted for D'_1, thus we can consider the drawings D'_2,…,D'_k to be uncrossed drawings for K_n-1 obtained from K_n by removing v.
Note that these uncrossed drawings of K_n-1 cover E(K_n-1).
Then, each D'_j contributes at most 2n-4 new uncrossed edges of K_n-1 as h(K_n-1) = 2(n-1)-2=2n-4 by <Ref>. 
So the number k of drawings satisfies

    n2≤ 2n-2 + (k-1)(2n-4) = (2n-4)k + 2.

However, 
(2n-4)k+2≤ (2n-3)k when k≥ 2, which is satisfied for n ≥ 5 by (<ref>).
Hence, for k' being the smallest positive integer that satisfies n2≤ k'(2n-3) we obtain k ≥ k'.
Thus, we again have the inequality (<ref>).




  
  
  

  
  
  

  
  
  
  
  
  








§ PROOF OF THEOREM 2



In this section, we prove <Ref> by providing the exact formula for the uncrossed number of complete bipartite graphs.
That is, we show

    (K_m,n)
    = 
      ⌈mn/2m+n-2⌉,    for  m ≤ n ≤ 2m-2 
    ⌈mn/2m+n-1⌉,    for  n=2m-1 
    ⌈mn/2m+n⌉,    for  6 ≤ 2m ≤ n 
    
        1,    for  m ≤ 2

for all positive integers m and n  with m ≤  n.
We start with the following useful lemma.



For positive integers m and n with m ≤ n, every outerplanar graph G that is a subgraph of K_m,n has at most 2m+n-2 edges.



Let G be an outerplanar graph that is a subgraph of K_m,n and let v and e be the number of vertices and edges of G, respectively. 
Since we are interested in the maximum number of edges of G, we can assume without loss of generality that G is connected and spanning in K_m,n.
Let f be the number of faces in an outerplanar drawing D of G.
By Euler's formula, we have v-e+f=2.
We also have 4(f-1)+2n ≤ 2e, because if we traverse the boundaries of all faces of D, then each edge is visited exactly twice. Moreover, each face, except for the outer face, gives a contribution of at least 4 edges.
For the outer face, the contribution is at least 2n since G as a subgraph of K_m,n is bipartite, all n vertices of the larger color class must be traversed, and the vertices of the outer face have to alternate between the color classes.

By subtracting Euler's formula 2 times from half of the inequality, we obtain 

    2(f-1)+n -2v+2e-2f ≤ e - 4,

which gives

    e ≤ 2v-n-2.

Since, v=m+n, we obtain e ≤ 2m+n-2.


We note that the bound from <Ref> is tight for all values of m and n.
The bound is attained for the ladder graph on 2m vertices with 3m-2 edges where we suitably append n-m leaves, obtaining an outerplanar subgraph of K_m,n with m+n vertices and 2m+n-2 edges; see Figure <ref>.




To obtain tight formulas for (K_m,n), we will need to also consider graphs that are not outerplanar.
A double cycle is a graph that is obtained from a cycle (where we also allow cycle on 2 vertices) by replacing each edge {u,v} with a copy of C_4 on vertices {u,v,x,y} and edges {u,x}, {u,y}, {v,x}, and {v,y} where x and y are new vertices; see <Ref>.
We call the vertices from the original cycle (u and v) black vertices and the new vertices (x and y) white vertices.
A double cycle with leaves is a double cycle where we attach new vertices of degree 1 to black vertices; see <Ref>.
These new vertices are also white.
Observe that a double cycle with leaves is a bipartite graph where the two color classes are formed by black and white vertices, respectively.





We start with two technical lemmas which will be used to prove the lower bounds.
For a graph G containing a cycle C, a path P of length two in G is called a double-chord of C if both ends of P lie on C, but the middle vertex of P is not on C.



Let G be a bipartite connected graph whose color classes are called black and white, C be a cycle of length at least 6 in G, and P_1 and P_2 be two paths of G satisfying all of the following assumptions:
0pt
  
  *  For i=1,2, P_i is a chord of C (that is, an edge with black and white ends on C) or a double-chord of C with black ends.
  
  *  If both P_1 and P_2 are double-chords with ends at distance two on C, then they both end in the same pair of vertices on C.
  
  *  If both P_1 and P_2 are disjoint double-chords with ends at distance four on C, then the length of C is not equal to 8.
  
  *  For i=1,2, denoting by x and y the ends of P_i on C, both subpaths of C with the ends x and y contain an internal vertex not incident to P_3-i.

Assume that D is a planar drawing of G, and that if both P_1 and P_2 are chords, then they are drawn in different regions of the subdrawing of C in D.

Then there is a pair of vertices of G of opposite colors which do not share a face of D.




Let the ends of P_1 and P_2 on C be x_1,x_2,x_3,x_4 in this clockwise cyclic order on C (regardless of whether they belong to P_1 or P_2), and note that we do not require these four vertices to be pairwise distinct.
Let Q_i denote the subpath of C from x_i to x_i+1 (where x_5=x_1) in the clockwise direction.
We consider first the case that one of the paths joins two consecutive vertices of x_1,x_2,x_3,x_4 in their cyclic order on C, say, P_1 with the ends x_1 and x_2 (same as Q_1). 
Then P_2 symmetrically has ends x_3 and x_4 (same as Q_3), and it may possibly happen that x_2=x_3 and/or x_4=x_1.

If both P_1 and P_2 are chords, then, from our specific assumption on the drawing D, the internal vertices of Q_1 are separated from the internal vertices of Q_3 (by the cycle of C∪ P_1∪ P_2 passing through P_1 and P_2), and since they include both black and white vertices, we get a pair of opposite colors with no common face in D.
So, up to symmetry, P_1 is a double-chord. By symmetry again, we may as well assume that Q_1 is not longer than Q_3 if P_2 is also a double-chord.
Then, using assumption (<ref>) if Q_3 is a double-chord, we have that Q_3 is of length at least 3.
Now, one of P_1 or Q_1 has an internal vertex y separated in D from the internal vertices of Q_3, and we can thus choose an internal vertex of Q_3 of color opposite to that of y and not sharing a face with y.

Thus, it remains to consider that, up to symmetry, P_1 has ends x_1 and x_3 and P_2 has ends x_2 and x_4, such that all x_1,x_2,x_3,x_4 are distinct.
In the drawing D, clearly, no internal vertex of Q_1 shares a common face with an internal vertex of Q_3, and likewise for Q_2 and Q_4. 
It now suffices to find such a vertex pair with opposite colors.

If C is of length 6, then the only case fulfilling the assumptions is that of P_1 and P_2 both being chords, and then the remaining two vertices on C indeed are of opposite colors and form the desired pair.
If C is of length at least 8, then, respecting assumption (<ref>), one of the paths, say Q_1, is of length at least 3 and thus contains internal vertices of both colors.
If Q_3 is of length at least 2, then we again get a desired pair of opposite colors. So, let Q_3 be a single edge of C.
If any one of Q_2 or Q_4 is also a single edge, we contradict assumption (<ref>).
Then both Q_2 and Q_4 are of length at least 2, and the neighbor of x_3 on Q_2 is of the opposite color to the neighbor of x_4 on Q_4, and we are again finished.



We now prove the following lemma, which describes the structure of an uncrossed drawing of K_m,n with 3 ≤ m < n if it contains many edges.
In fact, the statement of this result is stronger than what we need for the proof of <Ref>.



For integers m and n with 3 ≤ m < n and for a ∈{0,1}, let D' be an uncrossed subdrawing of K_m,n with at least 2m+n-a edges.
Let the color classes of K_m,n be called black and white such that we have m black vertices.

Then either D' contains a vertex that is adjacent to all black vertices, or n ≥ 2m-a and D' represents a subgraph of a double cycle with leaves.



We assume that no white vertex is adjacent to all black vertices as otherwise we are done.
The uncrossed subdrawing D' is planar by definition and, by <Ref>,
for every black vertex b and every white vertex w, there is a face of D' which is incident with vertices b and w.



Since D' is a drawing of a graph G' that is a subgraph of K_m,n and has 2m+n-1> 2m+n-2 edges, it follows from <Ref> that this graph is not outerplanar.
Thus, G' contains a K_2,3 or K_4 as a minor. However, if K_4 is a minor of a bipartite graph G', then G' also contains a minor resulting from K_4 by subdividing at least one edge, which in turn contains a K_2,3 minor, too.
Since K_2,3 contains two 4-cycles that share a path on three vertices, it follows that there are cycles C and C' in G' that share exactly one path P that contains at least three vertices and we also have |V(C) ∖ V(C')| ≥ 1 and |V(C') ∖ V(C)| ≥ 1.


Let Q be the set of vertices of G' that are not in V(C) ∪ V(C').
Now, we show that all vertices from Q are white.
Since m < n, there are more white vertices than black ones and thus it suffices to show that there are no vertices of different colors in Q.
Suppose for contradiction that there are black and white vertices in Q.
The cycles C and C' determine in D' three pairwise disjoint open regions ϱ_1, ϱ_2, and ϱ_3 in ℝ^2.
If any region ϱ_i contains a black vertex, then, since m<n and since any two vertices of different colors share a face of D', ϱ_i has to contain all white vertices from Q.
In particular, ϱ_i contains at least one black and at least one white vertex.
This, however, is impossible as then there is a vertex contained in one of the sets P, V(C) ∖ V(C'), or V(C') ∖ V(C) that does not share a face of D' with any of these two vertices in ϱ_i (here we used that P has at least 3 vertices and that |V(C')∖ V(C)| ≥ 1 and |V(C)∖ V(C')| ≥ 1).
It follows that each region ϱ_i contains only white vertices and thus all vertices in Q are white.

Now, we show that all black vertices lie on a single cycle in G'.
Since all vertices from Q are white, all black vertices lie in V(C) ∪ V(C').
Let P_1, P_2, and P_3 be the three paths that we obtain from C and C' by removing their two common vertices u and v of degree 3.
If each P_i contains a black vertex, then Q is empty as every white vertex in each of the regions ϱ_1, ϱ_2, and ϱ_3 does not share a face of D' with some of these black vertices.
Since m<n, each path P_i contains more white vertices than black, and then u and v are black and each path P_i has at least three vertices.
Since Q is empty and G' has only at most m+n+1< 2m+n-1 edges as m ≥ 3, there is an edge of G' that is not an edge of C nor C'.
Then, however, there is a white and black vertex that do not share a face in D' as can be shown by routine case analysis. 
This contradicts the assumptions and thus some P_i does not contain a black vertex.
Therefore, all black vertices lie on some cycle C” of G'.
We assume that C” is chosen such that the number of edges of G' which are chords of C” (have both ends on C”) is minimized.

Let R be the set of vertices of G' that are not contained in C”.
All vertices from R are white and all their neighbors are black and lie on C”. 
We will be finished with the main part if we show that every vertex of R has at most one neighbor on C”, or exactly two neighbors on C” which are at distance 2 along C”, and that there are no chords of C” in G'.
Indeed, then G' would be a subgraph of a double cycle with leaves, or there would be two vertices r_1,r_2∈ R having the same pair of neighbors on C” which is a contradiction via <Ref>.


Assume that some vertex r∈ R has two neighbors x and y at a distance at least 4 along C”, and recall that r is not adjacent to all black vertices.
If there were no chords of C” in G' and every vertex of R∖{r} had less than 2 neighbors on C”, then G' would have at most 2m+n-2 edges (2m on C” plus at most m-1 incident to r plus n-m-1 incident to the remaining vertices of R), a contradiction.
Therefore, there exists a chord of C” or a vertex r'∈ R∖{r} with two neighbors on C” to which, together with the double-chord of r, we may apply <Ref> to derive a contradiction to the assumptions; except in one specific case of C” of length 8 which has two “crossing” double-chords in G'.
In the latter exceptional case, we have m=4 and n=6, and since we assume at least 2m+n-1=13 edges, one of r or r' has another neighbor on C”, which by a simple analysis again leads to a pair of vertices of opposite colors not sharing any face in D'.

Hence, no vertex r∈ R has two neighbors at a distance at least 4 along C”.
This also implies that no vertex r∈ R has three or more neighbors in C”, since either r would be adjacent to all three black vertices of C”, or r would have two neighbors at a distance at least 4 along C”.
We are left with showing that there are no chords of C” in D'.
First, if all chords were drawn in D' in the same region of C” and no vertex of R had two neighbors on C”, the G' would again have at most 2m+n-2 edges since G' is bipartite (by <Ref>), a contradiction to the assumptions.
Otherwise, we have two chords P_1={f} and P_2={f'} drawn in different regions of C” in D', or a chord P_1={f} of C” and a double-chord P_2 formed by r∈ R with two neighbors on C”.
We again get a contradiction to the assumptions by applying <Ref> to P_1 and P_2, except in one case of r having both its neighbors adjacent to one end of f on C”. 
In the latter case, we “reroute” the cycle C” through r, and so decrease the number of chords, which contradicts our choice of C”.

Since all vertices in R have degree at most 2, the sum of their degrees is at most the number of edges that are not in C”, that is, 2m+n-a-2m=n-a.
Thus, 2|R| ≥ n-a.
Since |R| = (n+m) - 2m = n-m, we obtain 2(n-m) ≥ n-a, which can be rewritten as n ≥  2m-a.
This finishes the proof.










Subsequently, we turn our attention to the upper bounds for the proof of <Ref>.



For every n ≥ 2m and m ≥ 3 the edges of the complete bipartite graph K_m,n can be covered by ⌈mn/2m+n⌉ double cycles with leaves (not necessarily edge-disjoint).




Let the vertices of the partition class of size n be w_1, w_2, …, w_n and let the other m vertices be b_1, b_2, …, b_m. Because n ≥ 2m we can build double cycles with n-2m leaves (<Ref>) in which all b_i are the black vertices and all w_j are the white vertices. For this, we consider the vertices b_i and w_j, respectively, circularly ordered by their indices, as illustrated in <Ref>. 
Each double cycle with leaves in our cover will be defined by a sequence of (black) degrees (d_1, d_2, …, d_m) with d_i≥ 4 and ∑ d_i = 2m+n, and a starting index 1 ≤ s ≤ n in the following way.

The black vertices b_i occur in the given cyclic order on a double cycle, and for all 1 ≤ i ≤ m there are exactly d_i-4 white leaves to the vertex b_i. 
The white vertices w_j are assigned as neighbors to the black vertices on that double cycle with leaves such that vertex b_i is adjacent to vertices w_j with s_i≤ j < s_i+d_i where s_i = s + ∑_k<i (d_k - 2) and j is considered modulo n. In other words, we distribute the white neighbors according to their circular order such that vertex b_1 is adjacent to a consecutive block of d_1 white vertices starting at vertex w_s, and the blocks for the other black vertices are shifted such that each has a two-vertex overlap with the previous block.

Assume we have constructed one such double cycle with leaves given by the degree sequence (d_1, d_2, …, d_m) and the starting index s. Then we construct all remaining of the sought ⌈mn/2m+n⌉ double cycles in the following way. We successively increase the starting index by the degree of the first vertex s → s+d_1 and shift the degree sequence one to the left (d_1, d_2, …, d_m) → (d_2, d_3, …, d_m, d_1). 
This ensures that in the next double cycle every black vertex b_i is adjacent to a block of white vertices starting at the first white vertex in circular order that b_i was not yet adjacent to. Hence, in the union of the constructed double cycles, every black vertex has a continuous interval of adjacent white neighbors. It remains to construct a starting sequence (d_1, d_2, …, d_m) such that the sum of every ⌈mn/2m+n⌉ circularly consecutive degrees is at least n, which we do in the following.

Let ℓ = ⌈mn/2m+n⌉, which is the number of constructed double cycles, and a = 2m+n/m, which is the average degree in the degree sequence for the black vertices. Set the degrees to d_i = ⌊ i · a ⌋ - ∑_k<i d_k, which are integers and we have d_i≥4. Then the sum of the first ℓ degrees is ∑_k ≤ℓ d_k = ⌊ℓ· a ⌋ = ⌊⌈mn/2m+n⌉· a ⌋≥⌊mn/2m+n· a ⌋ = ⌊mn/2m+n·2m+n/m⌋ = n as intended.
So, if we extend the definition of the degrees for indices greater than m by d_i+m = d_i, we get that the sum of ℓ degrees starting at d_i+1 in the circular order of degrees is ∑_i < k ≤ℓ+i d_k = ⌊ (ℓ+i) · a ⌋ - ⌊ i · a ⌋≥⌊⌊ℓ· a ⌋ + ⌊ i · a ⌋⌋ - ⌊ i · a ⌋ = ⌊ℓ· a ⌋≥ n. Hence, all circular intervals of length ℓ sum up to at least n.




We need one more special case for which we use double cycles with one white vertex removed, that is, all black vertices have degree 4 except for two consecutive ones that only have degree 3.



For n = 2m-1 the edges of the complete bipartite graph K_m,n can be covered by ⌈mn/2m+n-1⌉ double cycles with one white vertex removed.



Since n = 2m-1, we aim to create ⌈mn/2m+n-1⌉ = ⌈m(2m-1)/2m+2m-1-1⌉ = ⌈m/2⌉ double cycles with one white vertex removed. In these graphs, every white vertex is adjacent to two consecutive black vertices in the circular order. We start with an arbitrary assignment of the black and white vertices for the first double cycle. For every further double cycle, we keep its basic structure and the assignment of the white vertices, but we cyclically shift the black vertices by two indices. That is, each black vertex is placed in every second position for black vertices in the process. Since we produce ⌈m/2⌉ double cycles with one white vertex removed in total, every black vertex is adjacent to every white vertex at least once. Hence, each edge of K_m,n is covered by at least one of the double cycles.



We are now ready to determine the uncrossed numbers of complete bipartite graphs K_m,n for all positive integers m and n.



Let m and n be positive integers with m ≤ n.
Recall that we want to prove that 

    (K_m,n)
    = 
      ⌈mn/2m+n-2⌉,    for  m ≤ n ≤ 2m-2 
    ⌈mn/2m+n-1⌉,    for  n=2m-1 
    ⌈mn/2m+n⌉,    for  6 ≤ 2m ≤ n 
    
        1,    for  m ≤ 2.


First, if m ≤ 2, then K_m,n is planar and we have (K_m,n) = 1.
Thus, we assume m ≥ 3 for the rest of the proof.

We now prove the upper bounds on (K_m,n).
By (<ref>), we have (K_m,n) ≤θ_o(K_m,n) and, since θ_o(K_m,n) = ⌈mn/2m+n-2⌉ for all m and n with m ≤ n by (<ref>), we have (K_m,n) ≤⌈mn/2m+n-2⌉.
This matches our desired formula for m ≤ n ≤ 2m-2, so it suffices to consider n ≥ 2m-1.
If n=2m-1, then the edges of K_m,n can be covered by ⌈mn/2m+n-1⌉ double cycles with one white vertex removed by Lemma <ref>.
If n ≥ 2m, then the edges of K_m,n can be covered by ⌈mn/2m+n⌉ double cycles with leaves by Lemma <ref>.
Observe that each double cycle C with leaves can be extended to a drawing of K_m,n where the edges of the double cycle are uncrossed as each white vertex shares a face in C with all black vertices.
Altogether, this gives the desired upper bounds on (K_m,n) in all cases.

It remains to verify the lower bounds. Recall that h(G) denotes the maximum number of uncrossed edges in any drawing of G.
We get (K_m,n) ≥⌈mn/h(K_m,n)⌉ from (<ref>). Since

    h(K_m,n) = 
    
        2m+n-2,    for  m=n 
    
        2m+n-1,    for  2m > n > m 
    
        2m+n,    for  n ≥ 2m,

by Theorem <ref>, we obtain the lower bound

    (K_m,n) ≥⌈mn/2m+n-2⌉,    for  m=n 
    ⌈mn/2m+n-1⌉,    for  2m > n > m 
    ⌈mn/2m+n⌉,    for  n ≥ 2m.

This matches our desired formula in all cases except for m<n≤ 2m-2.
Thus, we assume that m+1 ≤ n≤ 2m-2 from now on.
We want to show that (K_m,n) ≥⌈mn/2m+n-2⌉.
Let {D_1,…,D_k} be an uncrossed collection of drawings of K_m,n and let D'_1,…,D'_k be the corresponding uncrossed subdrawings of K_m,n.
If every drawing D'_i contains at most 2m+n-2 edges, then k ≥⌈mn/2m+n-2⌉.

Thus, suppose that some D'_i contains at least 2m+n-1 edges.
Without loss of generality, assume that it is the case for D'_1.
Then, D'_1 contains exactly 2m+n-1 edges as h(K_m,n) ≤ 2m+n-1 for m<n ≤ 2m-2.
We have 3 ≤ m<n and thus we can apply Lemma <ref>.
Since n≤ 2m-2, Lemma <ref> implies that D'_1  contains a vertex v that is adjacent to all m black vertices.
In every drawing D'_j with j > 1, the edges incident to v are already counted for D'_1, thus we can consider the drawings D'_2,…,D'_k to be uncrossed subdrawings for K_m,n-1 obtained from K_m,n by removing v.
Then, each D'_j contributes at most h(K_m,n-1) new uncrossed edges of K_m,n-1.

If each drawing from D'_2,…,D'_k contributes at most 2m+n-3 edges of K_m,n-1, then 

    mn ≤ (2m+n-1) + (k-1)(2m+n-3) = k(2m+n-3) + 2

as every edge of K_m,n is uncrossed in some drawing D'_i with i ∈{1,…,k}.
This can be rewritten as k ≥mn-2/2m+n-3.
We have mn-2/2m+n-3≥mn/2m+n-2 for m ≥ 4 and n ≥ m+2, and for m ≥ 5 and n=m+1, as the inequality can be equivalently expressed as n ≥ (4m-4)/(m-2).
Then, we get the desired bound k ≥⌈mn/2m+n-2⌉ as

    k ≥mn-2/2m+n-3≥mn/2m+n-2

and k is an integer.

Recall that we want to prove the lower bound k ≥⌈mn/2m+n-2⌉ in each case m +1 ≤ n ≤ 2m-2 where we know that D'_1 contains 2m+n-1 uncrossed edges.

If n=m+1, then Theorem <ref> gives h(K_m,n-1) =2m+(n-1)-2=2m+n-3 and we see that each drawing from D'_2,…,D'_k contributes at most 2m+n-3 edges of K_m,n-1.
We are then done by (<ref>) if m ≥ 5.
Otherwise, we have m=3,n=4 or m=4,n=5.
In both of these cases, the bound k ≥m^2+m-2/3m-2 implies that k ≥ 2, which also matches the upper bound on (K_m,n).

Finally, assume n ≥ m+2. 
Since we also have m ≤ 2m-2, we obtain m ≥ 4.
It follows from Theorem <ref> that h(K_m,n-1) =2m+(n-1)-1=2m+n-2 and we get 

    mn ≤ (2m+n-1) + (k-1)(2m+n-2) = k(2m+n-2) + 1,

which can be rewritten as k ≥ (mn-1)/(2m+n-2). 
Since k is an integer, this implies the desired bound k ≥⌈mn/2m+n-2⌉ if mn is not congruent to 1 modulo 2m+n-2.
Thus, we assume that mn = a(2m+n-2) + 1 for some positive integer a.
If every drawing from D'_2,…,D'_k contains at most 2m+n-3 uncrossed edges of K_m,n-1, then we are done by (<ref>) as m ≥ 4 and n ≥ m+2.
Otherwise, some drawing from D'_2,…,D'_k contains at least 2m+n-2 uncrossed edges of K_m,n-1.
Without loss of generality, this happens for D'_2.
Since n ≥ m+2, we obtain n-1 ≥ m+1 and Lemma <ref> implies that D'_2  contains a vertex v' that is adjacent to all m black vertices.

If n=m+2, then the drawings D'_2,…,D'_k to be uncrossed subdrawings for K_m,n-2 obtained from K_m,n by removing v and v' then contain at most 2m+(n-2)-2 = 2m+n-4 uncrossed edges of K_m,n-2 by Theorem <ref>.
Thus, we get 

    mn ≤ (2m+n-1) + (2m+n-2) + (k-2)(2m+n-4)
 implying
k ≥ (mn-5)/(2m+n-4).
We have k ≥ (mn-5)/(2m+n-4) ≥ (mn)/(2m+n-2) for m ≥ 3 and n ≥ 10(m-1)/(2m-5).
Thus, we are done in this case.
Otherwise, since m+2 = n, we have m+2 = n < 10(m-1)/(2m-5), which is satisfied only for m ∈{4,5} as m ≥ 4.
Since mn=a(2m+n-2)+1 and n=m+2, we either get 24 = 12a+1 if m=4 or 35 = 15a+1 if m=5, but none of these equations has an integer solution for a.
Thus, we indeed get the desired bound k ≥ mn/(2m+n-2).

If n ≥ m+3, then m+3 ≤ n ≤ 2m-2 implies m ≥ 5.
Theorem <ref> gives only

    mn ≤ (2m+n-1) + (2m+n-2) + (k-2)(2m+n-3)
 implying
k ≥ (mn-3)/(2m+n-3).
We have k ≥ (mn-3)/(2m+n-3) ≥ (mn)/(2m+n-2) for m ≥ 4 and n ≥ 6(m-1)/(m-3).
Thus, we are done in this case.
Otherwise, since m+3 ≤ n, we have m+3 ≤ n < 6(m-1)/(m-3), which is satisfied only for m ∈{5,6} as m ≥ 5.
Since mn=a(2m+n-2)+1, we either get 5n = a(n+8)+1 if m=5 or 6n = a(n+10)+1 if m=6, that is, either n=(8a+1)/(5-a) or n = (10a+1)/(6-a), respectively.
However, n is an integer satisfying m+3 ≤ n ≤ 2m-2, which gives n = 8 if m=5 or n ∈{9,10} if m=6, and no integer a satisfies (8a+1)/(5-a) = 8 nor (10a+1)/(6-a) ∈{9,10}.

Altogether, we always get k ≥⌈mn/2m+n-2⌉, implying (K_m,n) ≥⌈mn/2m+n-2⌉, which finishes the proof of Theorem <ref>.
 
  
  
  

















§ PROOF OF THEOREM 3


Here, we show that every connected graph G with n ≥ 3 vertices and m ≥ 0 edges satisfies 

    (G) ≥⌈m/f(n,m)⌉

where f(n,m) = (3n-5+√((3n-5)^2-4m))/2.


Let 𝒟(G) = {D_1,…,D_k} be an uncrossed collection of drawings of G.
For every i ∈ [k], let D'_i be a subdrawing of D_i containing only edges of D_i that are uncrossed in D_i.
By <Ref>, each drawing D'_i is then a plane graph with the property that every edge of G that is not an edge of D'_i is contained in a single face of D'_i.
Moreover, since G is connected, we can assume without loss of generality by this lemma that each D'_i represents a connected subgraph of G as to bound (G) from below it suffices to consider drawings D'_i with as many edges as possible.

Fix an arbitrary i ∈ [k].
The number of vertices of D'_i equals n.
We use m_i to denote the number of edges of D'_i and we will show that m_i ≤ f(n,m).

We set ℱ_i to be the set of faces of D'_i and f_i=|ℱ_i|.
For a face F of D'_i, we use v(F) for the number of vertices of D'_i that are contained in the boundary of F and we write |F| for the number of times we meet an interior of an edge of D'_i when traversing F along its boundary.
That is, |F| is the length of the facial walk.
Note that each edge can be counted once or twice in |F| and so we have v(F) ≤ |F| as D'_i represents a connected subgraph of G.
We assume that at least one edge of F is counted once in |F| and that v(F) ≥ 3 for every face F as otherwise there is only a single face in ℱ_i and D'_i is a tree with m_i ≤ n-1 ≤ f(n,m) for n ≥ 3. 
Also, observe that 

    ∑_F ∈ℱ_i |F| = 2m_i

as every edge is incident to a face of D'_i from the left and from the right.


Since every edge of G that is not an edge of D'_i is contained in a single face of D'_i, we have

    ∑_F ∈ℱ_i(v(F)2 - v(F)) ≥ m-m_i.

This is because vertices of each face F can span up to v(F)2 edges of D_i and at least v(F) pairs of vertices of D_i are already used for edges of D'_i as each face F contains an edge that is counted only once in |F|.
The left hand side of (<ref>) can be rewritten as 

    1/2∑_F ∈ℱ_i v(F)(v(F)-3).

Since v(F) ≥ 3 and |F| ≥ v(F) for every face F from ℱ_i, we obtain 

    1/2∑_F ∈ℱ_i |F|(|F|-3)≥ m-m_i.
 
Since |F|-3 ≥ 0, the left-hand side can be bounded from above by

    1/2(∑_F ∈ℱ_i |F|)(∑_F ∈ℱ_i (|F|-3)) = m_i(2m_i-3f_i)

where we used (<ref>) twice.
Altogether, we obtain m_i(2m_i-3f_i) ≥ m-m_i, which can be rewritten as

    f_i ≤2m_i/3 - m-m_i/3m_i.

Plugging this estimate into Euler's formula n-m_i+f_i=2, we get

    m_i ≤ 3n - 5 -m/m_i,

which after solving the corresponding quadratic inequality for m_i gives the final estimate

    m_i ≤ (3n-5+√((3n-5)^2-4m))/2 = f(n,m).


Since i was arbitrary, we see that each drawing D'_i contains at most f(n,m) edges of G and therefore, we indeed have 

    k ≥(G) ≥⌈m/f(n,m)⌉.











§ PROOF OF THEOREM 4





In this section, we prove that  is -complete.
Membership of this problem in the class  is trivial.
To show -hardness, we reduce from the following -complete problem  <cit.>.

Maximum Outerplanar SubgraphA graph G = (V,E) and a positive integer k.Is there an outerplanar subgraph of G with at least k edges?

Assume an instance of Maximum Outerplanar Subgraph.
Let M > |V|, say M=2|V|, and k'=|E|-k.
We augment G into a graph G', and show that G' can be drawn with at most M k'+|V| crossed edges, if and only if G admits an outerplanar subgraph with at least k edges.
The graph G' is obtained via two augmenting steps:
We add a central star, i.e., a vertex with an edge to each original vertex of G.
Then, we replace each original edge in G by M parallel paths of length two, which we call an M-bundle.
An example of this transformation can be seen in <ref>.

Suppose there is a drawing of G' with at most M k' + |V| crossed edges.
We want to modify this drawing into a drawing of G.
To this end, we first remove every path belonging to an M-bundle, if either of its two edges is crossed.
We also remove the central vertex and all of its incident edges.
All remaining edges are uncrossed and belong to an M-bundle path.
As there are at most M k' + |V| < M (k'+1) crossed edges in the drawing, for at least |E|-k'=k edges from G there is at least one path of its corresponding M-bundle that is not removed.
We contract for each edge of G one edge of one of the remaining paths of its M-bundle and remove all other M-bundle paths.
The vertices from G all share the face created by removing the central vertex and all vertices from M-bundles are either contracted or removed.
Thus, we have an outerplanar drawing of a subgraph of G with at least k edges.

Similarly, for every outerplanar subgraph H of G with at least k edges we can construct a drawing of G' with at most M k' + |V| crossed edges.
First, we draw H in an outerplanar embedding, then we draw the central star into the outer face.
Next, we draw the at most |E|-k = k' remaining edges of G in such a way that they only cross one another and the edges of the central star.
Finally, we replace every edge of G with an M-bundle.
The newly added vertices are positioned in such a way that at most one of the edges of each path is crossed.
Therefore, there are at most M k' crossed edges from the M-bundles and at most |V| crossed edges from the central star, for a total of at most M k' + |V| crossed edges.





§ PROOF OF THEOREM 6




We show that if Outerthickness is -hard, then  is -hard as well using a reduction from Outerthickness to .

The reduction employs similar arguments as used in <Ref>.
Let (G,k) be an instance of the problem Outerthickness.
We augment the input graph G into a graph G' by adding a vertex and connecting each vertex of G to it with a path of length two.
We call the added structure the central star.
See <ref> for an example of this transformation.

Let D be a drawing of G'.
Consider the uncrossed subdrawing D'_G consisting of the vertices and all uncrossed edges from G.
As there is a path in D∖ D'_G between each two vertices from G, we know that D'_G is outerplanar.
Thus, if (G') ≤ k and D_1,…,D_k is an uncrossed collection of G', then the respective subdrawings restricted to G decompose G into k outerplanar graphs.

Conversely, if G can be decomposed into k≥2 outerplanar subgraphs G_1, …, G_k, then we can construct an uncrossed collection D_1,…,D_k of G' in the following way:
In every drawing D_i, we first draw G_i as an outerplanar graph and we embed the central star in the outer face.
Then, we draw the remaining original edges in such a way that they only cross each other and edges from the central star.
In D_1, all crossings on the central star lie on edges incident to vertices of G, and in all other drawings, the crossings on the central star involve only edges incident to the universal vertex.
This way we assure that also every edge of the central star is uncrossed in some drawing.






§ CONCLUSIONS AND OPEN PROBLEMS


We provided exact values of the uncrossed number for complete and complete bipartite graphs.
The hypercube graphs form another natural graph class to consider as their outerthickness and thickness were determined exactly; see <cit.>.
However, we are not aware of any formula for the uncrossed number for the hypercube graphs.


Determine the exact value of the uncrossed number for the hypercube graphs.



In <Ref>, we determined a general lower bound on (G) in terms of the number of the edges and vertices of G by showing (G)≥⌈m/c n⌉ - O(n) - O(m) for some constant c with 0 < c ≤ 3.
In particular, we argued that the smallest obtainable constant c is approximately 2.82 for the case of dense n-vertex graphs with ε n^2 edges where ε∈ (0,1/2) is a fixed constant.
Can one obtain a better leading constant in the general lower bound on (G) for such dense graphs G?


We also propose investigating other properties of the uncrossed number. 
We conjecture that the uncrossed number can be arbitrarily far apart from the outerthickness despite them being quite similar on the graph classes we mainly investigated in this paper.
In fact, it follows from our results that the difference between the outerthickness and the uncrossed number of complete and complete bipartite graphs is never larger than one.


  For every positive integer k, there is a graph G such that 
  
    θ_o(G)-(G) ≥ k.




Lastly, it would be interesting to finally settle the computational complexity of the outerthickness problem.
We conjecture that the Outerthickness problem is -hard.
Note that if true, this would also settle the computational complexity of  by <Ref>.














 Paper Title: On the Eigenvalues of Graphs with Mixed Algebraic Structure.


[†]University of Groningen — Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence; Groningen, The Netherlands
GalleryGPT: Analyzing Paintings with Large Multimodal Models
    Heng Tao Shen
    Received 9th July 2024; accepted 13th July 2024
============================================================



We study some spectral properties of a matrix that is constructed as a combination of a Laplacian and an adjacency matrix of simple graphs. The matrix considered depends on a positive parameter, as such we consider the implications in different regimes of such a parameter, perturbative and beyond. Our main goal is to relate spectral properties to the graph's configuration, or to basic properties of the Laplacian and adjacency matrices. We explain the connections with dynamic networks and their stability properties, which lead us to state a conjecture for the signature.   









15A18, 15A15, 05C50, 05C70, 05C90, 47A55, 34D20




§ INTRODUCTION


The study of algebraic structures associated with graphs is remarkably relevant for pure and applied mathematics. In the first place, the study of matrices associated with networks can provide information on some features of the associated graph, and vice-versa <cit.>. In addition, in the realm of complex dynamical systems, the linear part of a vector field can be recognised as a matrix related to a network <cit.>. In particular, when studying the stability of the critical points of a vector field, one is especially interested in the eigenvalues of the linearisation at such points. In turn, for matrices associated with graphs, it is often possible, or desirable, to have some connection between the eigenvalues and some graph properties <cit.>. Consequently, we can relate the stability properties of complex dynamical systems to the properties of a graph.

A lot of work has been done to extend the possible algebraic structures associated with a graph, for example, accounting for directed edges or self-loops <cit.>, positive and negative weights <cit.>, hypergraphs <cit.>, etc. In this manuscript, we study graphs whose set of edges is split into two classes. We assign to one class a Laplacian structure and to the other class an adjacency structure. We propose a matrix, M(ϵ), that is a combination of such Laplacian and adjacency matrices, where the parameter ϵ acts as a global weight for the adjacency matrix. Precise definitions will follow in the text. Having in mind the potential application to gradient systems, or Hamiltonian, the Laplacian matrix represents stable interactions, while the adjacency matrix represents unstable saddle-like interactions, see Section <ref>. We exploit some relations between the graphs' properties and their spectrum. In particular, we characterise the perturbative regime to provide conditions for positive definiteness, which in turn can be related to the stability of the aforementioned dynamical systems. Additionally, we provide an upper bound for the number of transitions of the signature of M(ϵ) as the parameter ϵ varies. These transitions are important, among other reasons, due to their relationship to bifurcations in dynamical systems. Motivated by the intuition inferred from the dynamics, we state a conjecture for the monotonic change of signature, which we corroborate with a numerical investigation.

The paper is arranged as follows. First, in Section <ref> we introduce the main object of study together with some fundamental properties relevant for the treatment. Section <ref> is dedicated to perturbative results, while in Section <ref> we study the signature's transitions. Later, in Section <ref> we exploit the connection with networked dynamical systems. In Appendix <ref> we prove a result necessary for a proof (Proposition <ref>) contained in the main text.



§ PRELIMINARIES


Let ℱ={𝒱, ℰ} be a simple graph <cit.>, where 𝒱={ 1, …, n } is the set of vertices, and ℰ is the set of (undirected) edges {i,j}∈ℰ.


    The adjacency matrix, A_ℱ, of a graph  ℱ is the n× n matrix with components
    
    (A_ℱ)_ij :=
            
                1     if {i,j}∈ℰ , 
    
                0     otherwise .





    The Laplacian matrix, L_ℱ, of a graph  ℱ is the n× n matrix with components
    
    (L_ℱ)_ij :=
            (i)     if i=j , 
    
                -1      if {i,j}∈ℰ , 
    
                0     otherwise ,

    where (i) is the degree of the node i, i.e., the number of edges incident to the node i.




    Let us consider the subgraphs 𝒢={𝒱, ℰ_𝒢} and ℋ={𝒱, ℰ_ℋ} of ℱ, such that ℰ_𝒢∩ℰ_ℋ =∅ and ℰ_𝒢∪ℰ_ℋ = ℰ. We define the linear operator

    M(ϵ):= L_𝒢 + ϵ A_ℋ ,

where L_𝒢 is the Laplacian matrix of 𝒢, A_ℋ is the adjacency matrix of ℋ, and ϵ>0 is a positive real parameter.


Notice that ϵ acts as global weight on the adjacency matrix. In particular, in the limit ϵ=0 the matrix M(0) becomes the Laplacian matrix L_𝒢, while for ϵ≫ 1 the adjacency matrix A_ℋ will take the leading role. Such a structure can appear when modelling networked systems with different interactions, in that case the parameter ϵ can be seen as modulating the strength between the two coupling classes. For the entirety of the paper, otherwise explicitly mentioned, the notation relative to the graphs ℱ, 𝒢, ℋ,  and their definitions is given by Definition <ref>.











    Since L_𝒢 and A_ℋ are symmetric n × n matrices, then M(ϵ) is an n × n symmetric matrix. Therefore the eigenvalues μ_1(ϵ), …, μ_n(ϵ) of M(ϵ) are real, and there exists a complete basis of orthogonal eigenvectors.


We can immediately identify a couple of relevant properties of the matrix (<ref>). The first, is that the trace does not depend on the value of the parameter ϵ, but only on the number of edges of 𝒢.

    The trace of M(ϵ) is given by
    
    tr(M(ϵ))= 2 #ℰ_𝒢,

    where #ℰ_𝒢 denotes the number of edges of 𝒢.


    The statement follows straightforwardly from the fact that the adjacency matrix is traceless and that the trace of the Laplacian matrix is twice the number of edges of the associated graph. 


The other property we are going to state is related to the symmetries. We notice that the symmetry properties of the matrix (<ref>) are in correspondence with the symmetries of the graph ℱ, where the edges belong now to two classes associated respectively to 𝒢 and ℋ.


    A nonsingular matrix S is called a symmetry of a matrix M if S^-1MS=M, or equivalently [M,S]=0, where [·,·] denotes the commutator.

The set of matrices S that commute with the matrix M, together with the matrix product, form the symmetry group of M.


    Let ℱ be a simple graph. The automorphism group (ℱ) is the set of permutations leaving the adjacency structure of the graph ℱ invariant.


Let us consider the standard permutation-matrix representation of (ℱ) in ℝ^n, where n is the number of vertices of ℱ. Since this is the only representation we consider in this paper, we do not explicitly distinguish the group and its representation, so we write S ∈(ℱ). As a consequence of Definition <ref> we have that [S, A_ℱ]=0, ∀ S ∈(ℱ). Similarly, the commutation relations hold also for the Laplacian matrix.



    The group (𝒢) ∩(ℋ) is the symmetry group of the matrix M(ϵ).






 Let S ∈(𝒢) ∩(ℋ) then, by definition, we have [S, L_𝒢]=0, [S, A_ℋ]=0. Therefore [S,  L_𝒢 + ϵ A_ℋ] = [S,  L_𝒢 ]+ ϵ [S, A_ℋ]  =0, ∀ S ∈(𝒢) ∩(ℋ), and ∀ϵ.




    The matrix L_𝒢 is semi-positive definite, with ((L_𝒢)) equal to the number of connected components of 𝒢. Let 𝒢_1, …, 𝒢_r be the connected components of 𝒢, then the eigenvectors of the zero eigenvalues are 
    
    1_𝒢_1:=(1,⋯,1_#𝒱_𝒢_1, 0, ⋯, 0)^⊺, 1_𝒢_2:=(0, ⋯,0,1,⋯,1_#𝒱_𝒢_2, 0, ⋯, 0)^⊺, …, 1_𝒢_r:=(0, ⋯,0,1,⋯,1_#𝒱_𝒢_r)^⊺ ,

    where #𝒱_𝒢_i is the number of nodes in the connected component 𝒢_i, i=1, …, r.



    Let 𝒢 be a graph with r connected components. The normalised eigenvectors associated with the zero eigenvalues of L_𝒢 are given by
    
    1̂_𝒢_i:= 1_𝒢_i/√(#𝒱_𝒢_i) ,

    where i =1, …, r.



    The inertia of a matrix Q is the ordered triplet { N_-(Q) , N_0(Q) , N_+(Q) }, where N_-(Q) , N_0(Q) , N_+(Q) are respectively the number of negative, zero and positive eigenvalues of Q counted with multiplicity.



    The signature of a matrix Q is the number s(Q)=N_+(Q) - N_-(Q).



    For nonsingular symmetric matrices, the signature uniquely identifies the inertia and vice-versa. Moreover, thanks to Sylvester's law of inertia <cit.>, the signature and the inertia of a matrix are basis independent.











 §.§ Perturbation theory


In this section, we study the spectral properties of M(ϵ) in the regime of small perturbations, 0<ϵ≪1. Let us expand the eigenvalues of M(ϵ) in asymptotic series, 

    μ_i(ϵ) =  μ_i^(0) + ϵμ_i^(1) + ϵ^2  μ_i^(2) +   ,

where i =1, …, n. Notice that the zeroth-order terms are the eigenvalues of the Laplacian matrix L_𝒢, i.e., μ_i^(0)=λ_i, where λ_i ∈spec(L_𝒢). 


    Let 𝒢 be a graph with n vertices and r connected components. Then there are at least n-r eigenvalues of M(ϵ) that, for ϵ sufficiently small, are positive reals. 



    Since 𝒢 has r connected components, then L_𝒢 has r repeated zero eigenvalues and n-r positive eigenvalues. Thanks to Gershgorin Theorem <cit.>, for ϵ small enough the positive eigenvalues will remain positive.


Let us notice that Proposition <ref> holds for any small symmetric perturbation of L_𝒢. The situation is more delicate when we study the first-order perturbation of the zero eigenvalues of L_𝒢. In such case, it is relevant to consider the actual structure of the perturbation ϵ A_ℋ.



    Let 𝒢 be a graph with r connected components, 𝒢_1, …, 𝒢_r. Let μ_1(ϵ), …, μ_r(ϵ) be the eigenvalues of M(ϵ) with zeroth-order term equal to zero, i.e., μ_1^(0)= … =μ_r^(0)=0. The first-order corrections are μ_1^(1)= ϵθ_1, …, μ_r^(1)=ϵθ_r, where θ_1, …, θ_r are the eigenvalues of the matrix
    
    [             2#ℰ_ℋ^(𝒢_1)#𝒱_𝒢_1 #ℰ_ℋ^(𝒢_1,𝒢_2)√(#𝒱_𝒢_1#𝒱_𝒢_2)                             … #ℰ_ℋ^(𝒢_1,𝒢_r)√(#𝒱_𝒢_1#𝒱_𝒢_r); #ℰ_ℋ^(𝒢_2,𝒢_1)√(#𝒱_𝒢_2#𝒱_𝒢_1)             2#ℰ_ℋ^(𝒢_2)#𝒱_𝒢_2                             … #ℰ_ℋ^(𝒢_2,𝒢_r)√(#𝒱_𝒢_2#𝒱_𝒢_r);                             ⋮                             ⋮                             ⋱                             ⋮; #ℰ_ℋ^(𝒢_r,𝒢_1)√(#𝒱_𝒢_r#𝒱_𝒢_1) #ℰ_ℋ^(𝒢_r,𝒢_2)√(#𝒱_𝒢_r#𝒱_𝒢_2)                             …             2#ℰ_ℋ^(𝒢_r)#𝒱_𝒢_r ] ,

    where #ℰ_ℋ^(𝒢_i) is the number of edges of ℋ connecting nodes of 𝒢_i, and #ℰ_ℋ^(𝒢_i,𝒢_j) is the number of edges of ℋ connecting a node of 𝒢_i to a node of 𝒢_j.


    We need to find the first-order correction for the repeated zero eigenvalues of L_𝒢. Since the matrix L_𝒢 is symmetric we have r distinct orthogonal eigenvectors associated with the zero eigenvalues (<ref>). So, by applying the results of <cit.> for weak interactions, we find that the equation to be solved is 
    
    ( ⟨ A_ℋ1̂_𝒢_i, 1̂_𝒢_j⟩ - θδ_ij) ,

    where i,j = 1, …, r, δ_ij is the Kronecker delta, and ⟨· , ·⟩ is the standard scalar product in ℝ^n. By computing explicitly the matrix elements ⟨ A_ℋ1̂_𝒢_i, 1̂_𝒢_j⟩ we retrieve the matrix (<ref>).




    Let 𝒢 be a graph with two connected components, 𝒢_1 and 𝒢_2. For ϵ sufficently small, the matrix M(ϵ) is positive definite iff
    
    (#ℰ_ℋ^(𝒢_1,𝒢_2))^2< 4 #ℰ_ℋ^(𝒢_1)#ℰ_ℋ^(𝒢_2).

    



    For a graph 𝒢 with two connected components the matrix (<ref>) reduces to
    
    [            2#ℰ_ℋ^(𝒢_1)/#𝒱_𝒢_1 ℰ_ℋ^(𝒢_1,𝒢_2)/√(#𝒱_𝒢_1#𝒱_𝒢_2); ℰ_ℋ^(𝒢_2,𝒢_1)/√(#𝒱_𝒢_2#𝒱_𝒢_1)            2#ℰ_ℋ^(𝒢_2)/#𝒱_𝒢_2 ] =: 
            [ a c; c b ].

    The eigenvalues of (<ref>) are 
    
    θ_1,2 = a+b ±√((a-b)^2 + 4 c^2)/2 .

    Clearly, the first eigenvalue, θ_1= (a+b + √((a-b)^2 + 4 c^2))/2, is always positive as all the terms are positive. The second eigenvalue can be rewritten as θ_2= (a+b - √((a+b)^2 - 4 (ab- c^2)))/2, which is positive only if c^2<ab. The statement follows.
    




    Let 𝒢 be a connected graph and ℰ_ℋ non-empty, then, for ϵ sufficiently small, the matrix M(ϵ) is positive definite. Moreover, the first-order expansion of the zero eigenvalue of L_𝒢 is
    
    μ_0(ϵ) = ϵ2 #ℰ_ℋ/n + O(ϵ^2) .




    The statement follows from the explicit computation of a Rayleigh quotient <cit.>, or simply from the one-dimensional component case of Corollary <ref>.








 §.§ Bifurcation of eigenvalues


When the parameter ϵ is not restricted to small values, the study of the eigenvalues of M(ϵ) becomes more complicated. Indeed, by varying ϵ in the domain of positive reals, one can ask how many times the eigenvalues change sign. Such information is encoded in the inertia and in the signature of the matrix M(ϵ), see Definitions <ref>, <ref>. In the following proposition, we provide an upperbound on the changes of signs for the eigenvalues of M(ϵ).



    Let ϵ>0, and s(M(ϵ)) = N_+(M(ϵ)) - N_-(M(ϵ)) be the signature of the matrix M(ϵ). The maximum number of transitions of signature is
    
    n-((A_ℋ)) .




    The signature can change only if the matrix M(ϵ) is singular for some value of ϵ. In turn, M(ϵ) is singular if (M(ϵ))=0. In order to find an upper bound in the number of transitions of the signature, we need to study the maximal power in ϵ of the polynomial (M(ϵ)). Let M_ij(ϵ) be the components of the matrix M(ϵ), then the determinant can be written as
    
    
    
    
    (M(ϵ)) = ∑_σ∈𝕊_nsgn(σ) ∏_i^n (M(ϵ))_iσ(i) ,

    where 𝕊_n is the symmetric group of permutations over n elements, and sgn(σ) is the sign of the permutation σ. We recall that M(ϵ) is, by definition, M(ϵ)=L_𝒢 + ϵ A_ℋ. So, we have
    
    
    
    
    (M(ϵ)) = ∑_σ∈𝕊_nsgn(σ) ∏_i^n ( (L_𝒢)_iσ(i) + ϵ(A_ℋ)_iσ(i))  .

    We expand the product following the factorisation described in Appendix <ref>, obtaining 
    
    
    
    
    
    
    
    
    
    
    
    
    
    (L_𝒢) +    
        + ∑_σ∈𝕊_nsgn(σ) ( ∑_r=1^n-1ϵ^r C_n,r( (L_𝒢)_1σ(1), …,  (L_𝒢)_nσ(n);  (A_ℋ)_1σ(1), …, (A_ℋ)_nσ(n)) )  + 
       + ϵ^n  (A_ℋ) ,

    where C_n,r is defined in (<ref>), and roughly speaking it is the sum of the non-repeated combination of products where the terms (L_𝒢)_iσ(i) appear n-r times and  (A_ℋ)_jσ(j) appear r times, i≠ j.
    We recall that the Laplacian matrix of a simple graph is singular, so (L_𝒢)=0, and therefore one zero is always at ϵ=0. This fact is easy to check also without the above expansion. In order to have a non-zero term with power r in ϵ it is necessary that at least one term of the form
    
    (A_ℋ)_k_1σ(k_1)⋯(A_ℋ)_k_rσ(k_r) ,

    where k_1, … ,k_r are non-repeated combinations of elements in {1, …, n }, is non-zero. 
    
    
    
    
    For that to be true, the matrix A_ℋ needs to have r linearly independent eigenvectors. So, the maximum power of ϵ is bounded by the number of linearly independent eigenvectors of A_ℋ, which in turn is equal to the number of non-zero eigenvalues of A_ℋ, i.e., n- ((A_ℋ)). 
    



§ QUALITATIVE STUDY OF LINEAR VECTOR FIELDS


The treatment carried out in this paper is also related to the study of networked dynamical systems. Let us assign to each edge of the graph ℱ a potential function depending on the position coordinates associated to the connected vertices, i.e., {i,j}↦ V(q_i, q_j). The potential represents the interaction between the state variables defined on the vertex set. The total potential is given by

    V_ℱ := ∑_{i,j}∈ℰ V(q_i, q_j) .

We consider, for example, gradient vector fields

    q̇ = - ∇_q V_ℱ ,

where ∇_q:=(∂/∂_q_1, …, ∂/∂_q_n)^⊺ is a differential operator. 







The vector field (<ref>) inherits some properties and relations to the graph structure on which it is defined. Following the construction outlined at the beginning of the paper, Section <ref>, we split the interaction into two parts

    V(q_i, q_j) = 
        1/2 (q_i - q_j)^2     if {i,j}∈ℰ_𝒢 , 
    ϵ q_i q_j     if {i,j}∈ℰ_ℋ .


Now, if we explicitly compute equation (<ref>) we get

    q̇   = - M(ϵ) q,

where the matrix M(ϵ) is the one defined in (<ref>), which reads M(ϵ)=L_𝒢 + ϵ A_ℋ. Let us call the interactions of the form (q_i - q_j)^2/2 diffusive, while the interactions of the form q_i q_j we call saddle. Such terminology comes from the dynamical behaviour of the interactions when analysed separately. On the one hand, diffusive interactions give rise to the Laplacian matrix, that represent a linear diffusion process on the graph. For a connected graph, the system subject to diffusion converges to the mean value of the initial conditions. So, the interaction is stable and attracting, with a line of of equilibria representing the thermalisation values. Note that the line of equilibria is given by the eigenspace associated to the zero eigenvalue of the Laplacian. On the other hand, the term saddle is used to classify the behaviour of systems that have both positive and negative eigenvalues. Clearly, the adjacency matrix being traceless has positive and negative eigenvalues. The behaviour close to a saddle is generically unstable, as the only stable lines are given by the eigenspaces associated with the positive eigenvalues of the adjacency matrix.

An interesting and surprising consequence of Corollary <ref>, Lemma <ref>, and Corollary <ref> is that, under some conditions related to the graph structure, the addition of weak, i.e., ϵ≪ 1, saddle interactions can further stabilise the system. As a particular application of Lemma <ref>, we have the following:


 Let ℱ be a simple graph and {𝒢,ℋ} subgraphs of ℱ as defined in Section <ref>. Let 𝒢 have two connected components, namely 𝒢_1={𝒱_𝒢_1,ℰ_𝒢_1} and 𝒢_2={𝒱_𝒢_2,ℰ_𝒢_2} and ℰ_ℋ be non-empty. Then:
 
     
  * If 𝒢_1 or 𝒢_2 is a complete graph, then, for ϵ>0 sufficiently small, the gradient dynamics q̇ = - ∇_q V_ℱ, as defined above, is unstable. 
     
  * Let both 𝒢_i be non-complete graphs. Let s_i be the number of saddle interactions within vertices in 𝒢_i, i=1,2, and s be the number of saddle interactions between vertices in 𝒢_1 and 𝒢_2 (we recall that these saddle interactions are defined through the edge set of ℋ). If s^2<s_1s_2, then, for ϵ>0 sufficiently small, the gradient dynamics q̇ = - ∇_q V_ℱ, as defined above, is stable. 
 





    We emphasise that a similar result as in Proposition <ref> can be stated for more than two connected components, see Corollary <ref>.






Increasing ϵ leads to possible transitions of stability, see Proposition <ref>. Moreover, when ϵ increases the behaviour induced by the adjacency matrix will become more and more dominant. In turn, this would imply an increasing predominance of the unstable interactions associated to the negative eigenvalues of A_ℋ. Essentially, the dimension of the unstable eigenspace grows as ϵ increases. For such reasons we state the following conjecture.


    The signature of the matrix M(ϵ) is a monotonic decreasing function of ϵ.


We corroborate the statement of Conjecture <ref> by a numerical investigation, Figure <ref>. Moreover, we can observe that the conjecture can be extended to any matrix of the form A+ ϵ B, where A is a symmetric real positive semidefinite matrix, and B is a symmetric real matrix, Figure <ref>. Let us notice that if we assume the conjecture to be correct, we can find another bound for the number of transitions of the signature of M(ϵ), complementing the result of Proposition <ref>. In fact, for the limit ϵ→∞ the signature of M(ϵ) becomes the signature of the adjacency matrix A_ℋ, which we denote by s(A_ℋ). In the regime of ϵ≫ 1, we can consider the perturbation problem  A_ℋ + (1/ϵ) L_𝒢. Let us recall that A_ℋ can have some zero eigenvalues that under the perturbation of L_𝒢 will generically perturb to some nonzero real number. Since we look for an upper bound in the number of transitions, we should consider the case where all the zero eigenvalues become negative under perturbation. So, the upper bound is

    n- s(A_ℋ)+((A_ℋ)) .

Then, combining (<ref>) and (<ref>) we have that an upper bound for the number of transitions of signature is given by

    min{ n-((A_ℋ)) ,  n- s(A_ℋ)+((A_ℋ)) } .

The best bound is obtained by examining these two conditions

    if  2 ((A_ℋ)) > s(A_ℋ)   n-((A_ℋ)) ,
       if  2 ((A_ℋ)) < s(A_ℋ)   n- s(A_ℋ)+((A_ℋ)) ,

and, of course, if the equality 2 ((A_ℋ)) = s(A_ℋ) holds then they are both equal. In Figure <ref> we show a comparison between the number of transitions detected and the upper bounds we derived.










§ OUTLOOK


In this paper, we introduced the matrix (<ref>), that we called M(ϵ), representing a mixing between the Laplacian and Adjacency matrix. We studied how some spectral properties are related to the graph's configurations. In turn, we exploited the connection with networks of dynamical systems, in particular with gradient systems. Similarly, one can also derive a connection with Hamiltonian systems, where the stability properties are affected by the symplectic structure. Of course, there are possible generalisations of the definitions we stated. Potential directions include considering other graph structures, e.g., directed graphs, or hypergraphs, but also other combinations of known algebraic structures. It might also be possible to explore in more detail other properties of the matrix M(ϵ), such as its eigenvectors. Moreover, it would be quite significant to prove or further understand Conjecture <ref>.





 












§ FACTORISATION OF THE PRODUCT ∏_I (X_I + Ε Y_I) 


We reserve this appendix for the factorisation in powers of ϵ of the product

    ∏_i=1^n (x_i + ϵ y_i).

The factorisation turns out suitable for the computation of the determinant of M(ϵ), see Section <ref>. In general, the expansion proposed provide the standard polynomial-in-ϵ form for the product (<ref>).


    The product (<ref>) can be rewritten as a polynomial of degree n in ϵ of the form
    
    ∏_i=1^n (x_i + ϵ y_i) = ∑_r=0^n ϵ^r C_n,r(x_1, …, x_n; y_1, …, y_n) ,

    where C_n,r(x_1, …, x_n; y_1, …, y_n) is defined as
    
    C_n,r(x_1, …, x_n; y_1, …, y_n):=  x_1 ⋯ x_n-r y_n-r+1⋯ y_n + … + y_1 ⋯ y_r x_r+1⋯ x_n_nr possible combinations .

    



    We start by expanding (<ref>) as follows
    
    ∏_i=1^n (x_i + ϵ y_i)    = ∑_k_1, …, k_n =0^1 ∏_i=1^n x^k_i (ϵ y)^1-k_i
       = ∑_k_1, …, k_n =0^1 ϵ^n- ∑_i=1^n k_i∏_i=1^n x^k_i y^1-k_i .

    Notice that the exponents k_i and 1-k_i are combinations of 0 and 1. So the sum ∑_k_1, …, k_n =0^1 gives the 2^n combinations of products ∏_i=1^n x^k_i y^1-k_i. We group together the combinations that have the same power of ϵ, i.e., n- ∑_i=1^n k_i =r, r=0,…, n. For a given r we have that there are nr products that are non-repeated combinations of n-r elements of { x_1, …, x_n} and r elements of  { y_1, …, y_n}.















abbrv





§ STATEMENTS AND DECLARATIONS



    
  Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
    
  Competing Interests The authors have no relevant financial or non-financial interests to disclose.
    
  Data Availability All datasets are available upon request to the corresponding author.





 Paper Title: Algorithmic Pot Generation: Algorithms for the Flexible-Tile Model of DNA Self-Assembly.





















































 



















definition
definitionDefinition
lemmaLemma
corollaryCorollary
theoremTheorem

proposition[theorem]Proposition
notation[theorem]Notation
question[theorem]Open Question

remark
*remarkRemark
example[theorem]Example

plain
conjectureConjecture[section]


 








amatrix[1]
	([ )



Journal of Molecular Graphics and Modelling















































ICERM,Rose]Jacob Ashworth
ICERM,Tufts]Luca Grossmann
ICERM,Johns Hopkins University]Fausto Navarrocor1
ICERM,Stonehill]Leyda Almodovar
ICERM,Lewis]Amanda Harsy
ICERM,CSU]Cory Johnson
ICERM,Converse]Jessica Sorrells



[cor1]Corresponding author. Email: fnavarro0806@gmail.com



[ICERM]organization=Institute for Computational and Experimental Research in Mathematics,
            addressline=121 S Main St, 
            city=Providence,
            postcode=02903, 
            state=RI,
            country=USA

[Rose]organization=Department of Mathematics, Rose-Hulman Institute of Technology,
             addressline=5500 Wabash Ave,
             city=Terre Haute,
             postcode=47803,
             state=IN,
             country=USA

[Tufts]organization=Department of Mathematics, Tufts University,
             addressline=419 Boston Ave,
             city=Medford,
             postcode=02155,
             state=MA,
             country=USA

[Hopkins]organization=Department of Mathematics, Johns Hopkins University,
             addressline=555 Pennsylvania Ave,
             city=Baltimore,
             postcode=21218,
             state=MD,
             country=USA

[Converse]organization=Department of Mathematics, Converse University,
             addressline=580 E. Main Street,
             city=Spartanburg,
             postcode=29302,
             state=SC,
             country=USA

[Stonehill]organization=Department of Mathematics, Stonehill College,
             addressline=320 Washington St,
             city=North Easton,
             postcode=02357,
             state=MA,
             country=USA

[Lewis]organization=Department of Mathematics, Lewis University,
             addressline=1 University Pkwy,
             city=Romeoville,
             postcode=60446,
             state=IL,
             country=USA

[CSU]organization=Department of Mathematics, California State University - San Bernardino,
             addressline=5500 University Pkwy,
             city=San Bernardino,
             postcode=92407,
             state=CA,
             country=USA




§ ABSTRACT 


Recent advancements in microbiology have motivated the study of the production of nanostructures with applications such as biomedical computing and molecular robotics. One way to construct these structures is to construct branched DNA molecules that bond to each other at complementary cohesive ends. One practical question is: given a target nanostructure, what is the optimal set of DNA molecules that assemble such a structure? We use a flexible-tile graph theoretic model to develop several algorithmic approaches, including a integer programming approach. These approaches take a target undirected graph as an input and output an optimal collection of component building blocks to construct the desired structure.






    < g r a p h i c s >







  * Integer programming solves the Subgraph Realization Problem

  * Integer programming finds optimal assembly designs for a given structure

  * Many new optimal assembly designs are computationally generated and verified





DNA self-assembly DNA tiles tile-based assembly branched junction molecules computational complexity spectrum of a pot nanotube lattice graph


[2020] 92E10 05C90 05C85 92D20











§ INTRODUCTION


Recent advancements in microbiology have motivated the study of the production of nanostructures with applications from drug delivery to molecular robotics and biomedical computing <cit.>.
As these structures become smaller and more complex, production becomes increasingly difficult. As a result, laboratories exploit the Watson-Crick complementary base-pairing properties of DNA strands to create synthetic DNA sub-components with the ability to self-assemble into a target structure <cit.>. Determining which molecules will construct a target DNA nanostructure is a critical step in the production of many microbiological tools. 

One prominent mathematical model of DNA self-assembly is referred to as the flexible-tile model, which was first introduced in <cit.>. This model refers to geometrically unrestrained abstractions of branched-junction DNA molecules as “tiles." Branched junction molecules are star-shaped, consisting of several arms with `cohesive-ends' that can only bond with complementary cohesive-ends, representing complementary sequences of base pairs (see Figure <ref>). In <cit.>, the flexible-tile model is further developed as a graph theoretical problem in which each vertex in a graph represents a branched-junction molecule and each edge is a pair of complementary cohesive-ends that have bonded. The flexible-tile model creates a framework in which to explore two design questions pertinent to laboratories: (1) Given a desired target nanostructure, how many distinct molecule types are required for self-assembly of the target? (2) Given a collection of molecule types, what types of nanostructures can be self-assembled from them?  In <cit.> Ellis-Monaghan et. al. also introduce three “scenarios," representing increasingly constrained levels of laboratory requirements, in which to consider these questions. These scenarios categorize the extent to which creation of byproducts (i.e. structures different from the target structure) are tolerated in the self-assembly process. We recap the relevant mathematical details of the model in 
Section <ref>.

In this paper, we develop algorithms to find the optimal multi-set of branched-junction molecules realizing a target DNA nanostructure. This requires engagement with two main combinatorial problems, both of which were shown to be NP-hard in <cit.>. 

The first problem is to determine whether a chosen set of branched-junction molecules could self-assemble into a nanostructure of a smaller order than the target structure. This is known as the Subgraph Realization Problem (SRP). A previous algorithm introduced in <cit.> addresses the SRP in a limited range of cases. Motivated by that work, we formulate the SRP as an integer program In Section <ref>, allowing us to solve it for arbitrary numbers of molecules under feasible computational constraints. Our integer programming based approach is capable of finding minimal tile sets for arbitrary pots with dozens of tiles and bond-edge types in less than a tenth of a second (see Table <ref>).

The second problem, which we refer to as the Optimal Pot Problem (OPP), is to determine whether a given set of molecules assembles into a graph. We construct another integer program that achieves this goal in Section <ref>. We use this integer program in Section <ref> to solve the problem in the “scenario" where all byproducts are tolerated. We then combine this integer program with the SRP integer program to address a more difficult version of the problem in which no byproduct structures of smaller size than the target structure are permitted to form from the proposed set of molecules in Section <ref>. As a result, we are able to computationally determine minimal sets of tiles, i.e. abstract representations of molecules, for a wide variety of graphs in two of the three “scenarios" of restriction described in <cit.>.

The algorithms presented in this paper (shown in Figure <ref>) can serve not only as a tool for laboratories seeking to produce target graph structures, but also for researchers looking to generalize patterns in families of graphs. Section <ref> discusses the implementation details, including verification of known optimal solutions and reporting of new solutions found by the algorithm for a variety of graphs.





§ METHODS





 §.§ Graph Theoretical Formalism


Throughout the paper, we will denote the set of nonnegative integers as ^+ and the set of nonnegative integers less than or equal to k as _k.

A discrete graph G consists of a set V=V(G) of vertices and a set E=E(G) of edges together with a map μ:E → V^(2)  where V^(2) is the set of (not necessarily distinct) unordered pairs of elements of V. If μ(e)={u,v}, then u and v are the vertices incident with edge e. We allow graphs to have loops and multiple edges, so that it is possible for μ(e) = {u,v} with u=v (in the case of a loop edge) or for μ(e)=μ(f) (in the case of multiple edges). We use the notation # to denote size or quantity, so that e.g. #V(G) will denote the order of G.

We denote a half-edge of a vertex v as (v,e) if v ∈μ(e). H shall denote the set of half edges of G. 

For vertex v_i, we denote its degree (the number of incident edges to v_i) as d_i. For a graph G, we denote the maximum vertex degree as Δ(G) and the minimum vertex degree in G as δ(G). We denote the neighborhood of vertex v, the set of all vertices adjacent to v, as N_G(v). 

When two graphs G_1 and G_2 are isomorphic, we denote this as G_1 ≅ G_2.



 §.§ Graph Assembly Model
 

To model the process of DNA self-assembly, we employ the graph-theoretical formalism described in  <cit.>. For the convenience of the reader, we include some of the relevant definitions in this section and introduce several new convenient tools for our purposes.

The k-armed branched junction molecule is the basic unit of this model. A k-armed branched junction molecule is a star-shaped molecule whose arms are formed from strands of DNA (see left image in Figure <ref>). At the end of each of these arms is a region of unpaired bases, forming a cohesive-end (also known informally as a sticky end). We encode cohesive ends using symbols.



  A cohesive-end type is an element of a finite set S=Σ∪Σ̂, where Σ is the set of hatted symbols and Σ̂ is the set of un-hatted symbols.  Each cohesive-end type corresponds to a distinct arrangement of bases forming a cohesive-end on the end of a branched junction molecule arm, such that a hatted and an un-hatted symbol, say a and â, correspond to complementary cohesive-ends.  We do not allow palindromic cohesive-ends,  so a cohesive-end corresponding to a is complementary to but different from the cohesive-end corresponding to â for all a∈Σ. Moreover we use the convention that â̂ =a.  

 A cohesive-end type joined to its complement forms a bond-edge type, which we identify by the un-hatted symbol. That is, Σ represents the set of all bond-edge types. 

 Cohesive-end types a and â may join to form a bond-edge of type a. This process is shown in Figure <ref>. 

Arms with complementary cohesive-ends can bond via Watson-Crick base pairing. In the simplest setting, the arms of a branched junction molecule are double stranded DNA with one strand extending beyond the other to form a cohesive region (see <cit.>). In this case, the molecules are quite flexible. The mathematical model we employ here assumes there are no geometric restrictions, for example on the inter-arm angles or arm lengths. See <cit.> for a model that encompasses such geometric constraints.

In this graph theoretical framework, we encode a k-armed branched junction molecule and its un-bonded cohesive ends as follows.

 A k-armed branched junction molecule is represented by a vertex of degree k with k incident half-edges called a tile. Thus,  a tile type t is a classification of a flexible-armed branched junction molecule according to its set of cohesive-end types. The half-edges of a tile are labeled by the cohesive-end types corresponding to the cohesive-ends on the arms of the molecule the tile represents. The tile type of a tile is thus given by the multi-set of its cohesive-end types whose multiple entries of the same cohesive-end type are indicated by the exponent to the corresponding symbol.  

For simplicity of computation, throughout this paper we often write a tile as a lexicographically ordered string.

 A tile of degree 3 with cohesive ends denoted as a, a, and b̂ is pictured in Figure <ref>. This tile would typically be written as {a^2,b̂}, but here we use aaB, where B is used in place of b̂.


Tiles as we have defined them here are sometimes referred to as `flexible tiles' to avoid conflation with rigid tiles seen in Wang tilings or tile assembly models (TAM) (see <cit.>). Note that a self-assembled structure may contain several tiles of the same tile type. For greater mathematical simplicity, we assume that once a lab creates a particular tile, an unlimited number of tiles of that type is available to be used in the self-assembly process. To these ends, we define a working collection of tiles as the set of distinct tile types it contains.


 A pot is a collection of tiles. More precisely, a pot is given by its collection of tile types, and this is represented by a set P = {t_1,...,t_k} where each t_i is a tile type for some tile in P, and for all a∈Σ, if there is i such that a ∈ t_i then there is j ∈{1,...,k} such that â∈ t_j. We arbitrarily choose an ordering of the tiles, although all orderings represent the same pot. The set of bond-edge types that appear in tile types of P is denoted with Σ(P), and we write #P to denote the number of distinct tile types in P and #Σ(P) to denote the number of distinct bond-edge types that appear in P. 

 An assembly design of a graph G is a labeling λ : G →Σ∪Σ̂ of the half-edges of G with the elements of Σ and Σ̂ such that if e ∈ E(G) and μ(e) = {u,v}, then λ((v,e)) = λ((u,e)). This means that each edge of G receives both a hatted and an un-hatted version  of the same symbol, one on each of its half-edges.  

 Given an assembly design λ, the labelled degree d_σ_i(v) denotes the number of half edges incident to vertex v which are labelled with cohesive-end type σ_i (i.e. such that λ((v,e)) = σ_i ∈Σ∪Σ̂) .


 An assembling pot P_λ(G) for a graph G with assembly design λ is the set P_λ(G) = {t_v | v ∈ V(G)} where t_v ={λ (v,e) | v ∈μ(e), e ∈ E(G)}.

 We say a pot P realizes the graph G if there exists an assembly design λ : H →Σ∪Σ̂ such that P_λ(G) ⊆ P.   

 The set of graphs with their associated assembly designs realized by a pot P, namely {(G,λ): P_λ(G) ⊆ P }, is called the output of P and is denoted by 𝒪(P). 


    Given an an assembly design λ, its realization function f: V(G) → P_λ(G) maps v to the tile type used to label v, i.e.
    
    f(v) = {σ_i^d_σ_i(v) : i ∈_N}∪{σ̂_i^d_σ_i(v) : i ∈_N},

    where d_σ_i(v) denotes the number of half edges incident to v which are labelled σ_i (see Definition <ref>). 


  Figure <ref> shows an assembly design for K_3. Its realization function can be defined as f(v_1) = {a,b}, f(v_2)={a, b̂}, f(v_3)={â^2}. 

    


Within the flexible tile model of DNA self-assembly, we can consider the strategic design of pots that can assemble given graphs. An overarching goal is to find pots with the fewest numbers of tile and/or bond-edge types, and thus to theoretically determine the minimum number of branched junction molecules needed for self-assembly of certain DNA nanostructures. Given a target graph G, we seek minimal pots P under consideration of three levels of restriction, which are the “scenarios” described in <cit.>. 

    
  * Scenario 1. Least Restrictive: G ∈𝒪(P). Note: This allows the possibility that there exists H ∈𝒪(P) such that #V(H) < #V(G). 
    
  * Scenario 2. Moderately Restrictive: G ∈𝒪(P) and for all H ∈𝒪(P), #V(H)≥#V(G). Note: This allows the possibility that there exists H ∈𝒪(P) such that #V(H)=#V(G) but H ≇G. 
    
  * Scenario 3. Highly Restrictive: G ∈𝒪(P), and for all H ∈𝒪(P), #V(H)≥#V(G), and if #V(H)=#V(G) then H ≅ G.
 


    Consider the different ways of assembling the graph K_4 shown in Figure <ref>. The left column shows a specific pot and how it realizes (self-assembles) K_4. This pot uses only one bond-edge type and is satisfactory in Scenario 1. At the bottom of the column is another, smaller order graph also realized by the same pot, which disqualifies the pot from Scenario 2. The middle column shows how another pot with three bond-edge types realizes K_4. The pot does not assemble into any smaller order graphs, so it is satisfactory for Scenario 2. At the bottom of the middle column is a graph non-isomorphic to K_4 with equal order to K_4, which disqualifies this pot from Scenario 3. Finally, the right column shows a realization of K_4 using a third pot. This pot cannot realize any graphs of smaller order or of equal order but non-isomorphic to K_4, so this pot is valid in Scenario 3.
    



Given the three theoretical laboratory scenarios, we can search for pots in each scenario that consist of the fewest tile types or fewest bond-edge types. This is analogous to seeking the smallest collections of molecules that will self-assemble into a given target nanostructure while also potentially avoiding byproduct structures, or waste. The following definitions provide notation for minimality with respect to tile types and bond-edge types.



T_i(G) = min{#P | P  realizes  G  according to Scenario  i}.



B_i(G) = min{#Σ (P) | P  realizes  G  according to Scenario  i}.


This paper focuses on finding optimal pots in Scenarios 1 and 2. This amounts to solving the following problem: 

Optimal Pot Problem (OPP) Given a graph G and scenario S_i, find pots with T_i(G) tile types and/or B_i(G) bond-edge types. 


    It was shown in <cit.> that it is always possible to find a pot simultaneously achieving T_1(G) and B_1(G), but that graphs exist such that no pot will simultaneously achieve T_3(G) and B_3(G). It was conjectured that there existed no pot realizing the cube graph and simultaneously achieving T_2(G) and B_2(G), but this was disproved in <cit.>. The term “biminimal pot" was coined in <cit.> to describe pots simultaneously achieving a minimum number of tile types and bond-edge types. The question of whether a biminimal pot satisfying Scenario 2 exists for every graph remained an open question until Toby Anderson, Olivia Greinke, Iskandar Nazhar, Luis Santori,   discovered that the Y_3,3 stacked prism graph does not have a biminimal Scenario 2 pot (pending citation). Thus, for Scenario 2 we present an algorithm to search for pots that are minimal with respect to either the number of tile types and the number of bond-edge types, with the knowledge that there might exist two distinct pots achieving these values.


From Scenario 2 (and Scenario 3) arises two problems that must be grappled with when attempting to design an assembling pot: (1) determining if a given pot will realize a graph of a given order, (2) determining whether a pot that realizes a given graph can also realize any smaller graph. These two problems and their computational complexity were studied in <cit.>. Below we reproduce the formulations of these problems.

Pot Realization Problem (PRP)
 Given a pot P and a positive integer k, does P realize a graph G with k vertices? 


Substructure Realization Problem (SRP)
 Given an assembling pot P_λ(H) for a graph H of order n and an integer ℓ with 0<ℓ<n, does P_λ(H) realize a graph F with ℓ vertices? 




 §.§ SRP Integer Program


Recall that in order to determine whether a pot P is valid for a given target graph G in Scenarios 2 or 3,  it must be known whether any graphs of order smaller than #V(G) are in 𝒪(P). To begin, we follow <cit.> and <cit.> to describe the original framework for the SRP and the challenges presented by certain cases. In <cit.> the augmented matrix of a linear system encoding some constraints on an assembling pot P_λ(G) for a given graph G is introduced as a first-line computational approach for determining whether P is valid in Scenario 2. 

The following definitions and propositions relating to the construction matrix are given in <cit.>.

 Let P = { t_1,...,t_p } be a pot and let z_i,j denote the net number of cohesive-ends of type a_i on tile t_j, where un-hatted cohesive-ends are counted positively and hatted cohesive-ends are counted negatively. Then the following system of equations must be satisfied by any connected graph in 𝒪(P):

    z_1,1r_1+z_1,2r_2+...+z_1,pr_p    =    0 
       ⋮   
    
    z_m,1r_1+z_m,2r_2+...+z_m,pr_p    =    0 
    
    r_1 + r_2 + ... + r_p    =    1


The construction matrix of P, denoted M(P), is the corresponding augmented matrix:


    M(P) = [ [ z_1,1 z_1,2     … z_1,p     0;     ⋮     ⋮     ⋱     ⋮     ⋮; z_m,1 z_m,2     … z_m,p     0;     1     1     …     1     1;       ] ].



 The solution space of the construction matrix of a pot P is called the spectrum of P, and is denoted  𝒮(P). 

The construction matrix is used primarily to determine whether there exists in 𝒪(P) a graph of smaller order than the desired target graph. For a given solution ⟨ r_1,...,r_p ⟩ to M(P), the least common denominator of the r_i's gives the smallest order of a graph in 𝒪(P). This is detailed in the following result from <cit.>.


Let P={t_1, … , t_p} be a pot.  Then:


    
  * If a graph G of order n is realized by P using R_j tiles of type t_j, then 1/n⟨ R_1, …, R_p⟩ is a solution of the construction matrix M(P), i.e. is in 𝒮(P).
    
  * If ⟨ r_1, …, r_p ⟩∈𝒮(P), and n is a positive integer such that nr_j ∈ℤ_≥ 0 for all j, then there is a graph of order n in 𝒪(P) that is realized by using nr_j tiles of type t_j. 
    
  * The smallest order of a graph in 𝒪(P) is min {lcm {b_j|r_j ≠ 0  and  r_j = a_j/b_j}} where the minimum is taken over all ⟨ r_1,…,r_p⟩∈𝒮(P) such that r_j ≥ 0 and a_j/b_j is in reduced form for all j.











In the cases where 𝒮(P) consists of a unique solution, the construction matrix quickly provides a solution to the SRP. However, if the system of equations defined by the construction matrix does not have a unique solution, this approach does not give an immediate answer to the SRP <cit.>. The spectrum 𝒮(P) will have degrees of freedom whenever #P > #Σ(P) + 1, as then the dimensions of M(P) will be (#Σ(P) + 1) ×#P.  This occurs even in examples of quite simple target graphs and small pots, as we demonstrate in the following example.

 The pot P = {{a},{â},{a,â}} realizes the graph P_3, the path graph of order 3. Since #P = 3 > 2 = #Σ(P) + 1 (i.e. M(P), shown below, has two more columns than rows), the system has one degree of freedom. 

    
    
    
    
    
    

    
    M(P) = 
        [  1 -1  0  0;  1  1  1  1 ]


    It is easy to verify,

    
    𝒮(P) = {1/2r⟨ r-t, r-t, 2t ⟩|   r ∈ℤ^+,  t ∈(ℤ∩ [0,r]) }.

    
    Here, r is a scaling factor and t represents the free variable. In this instance, there exist multiple H ∈𝒪(P) such that #V(H) < #V(G).  A graph of order one, a self-loop with the tile {a,â}, can be realized with the solution corresponding to r=t. The path graph P_2 can be realized (using tiles {a} and {â}) with the solution corresponding to t=0. Note that the target graph, P_3, is realized in accordance with r=3, t=1. 


 As can be observed in Example <ref>, when 𝒮(P) has degrees of freedom, one must check all valid values of the free variable(s) in order to determine if there exist smaller order graphs in 𝒪(P). In cases of larger graphs and pots, this becomes tedious and time-consuming. In addition to these difficulties, it was shown in <cit.> that an optimal pot in Scenario 2 might correspond to multiple distinct solutions (arising from distinct values of the free variable(s)) of M(P).  

In <cit.>, an hmhm which checks all possible tile distributions with non-negative integer tile quantities was proposed to handle cases where 𝒮(P) has degrees of freedom. In theory, the algorithm can solve the SRP for systems with any number of free variables. However, in cases with more than two free variables, the algorithm presents inhibitory runtime costs. Thus, the algorithm is only implemented for pots with one or two free variables. 

Instead, we reframe the SRP as an integer program. By using this program, we can efficiently determine the smallest order graph realized by a given pot without trying to find a solution to the construction matrix, allowing us to solve the SRP no matter how many free variables the spectrum of the pot has. 

Notice that the R_j in the first part of Theorem <ref> describe the number of tiles used in a specific assembly design of G.


    Given a pot P, a tile distribution is a tuple Ω(P) = (R_1, ⋯ R_#P), where R_i represents the number of tiles of type t_i.
 

Note that if Ω(P) describes the multi-set of tiles used to build a graph G, then 
    ∑_i=1^#PR_i = #V(G).



    Let Ω(P) = (R_1,...,R_#P) be a tile distribution of a given pot P = {t_1, ... t_p}. Then Ω(P) corresponds to some ⟨ r_1,...,r_p⟩∈𝒮(P) such that
    
    r_i = R_i/∑_k=1^#PR_k.






    The tile distribution associated with the realization of C_4 by the pot P = {aa, âb̂, bb} as shown in Figure <ref> is Ω(P) = (1, 2, 1).


Using tile distributions, the SRP can be reformulated as follows:

Substructure Realization Problem (SRP) Given a graph G of order N and a pot P which realizes G, does P realize a graph of order less than N with some tile distribution Ω(P)? 


The following lemma presents the SRP in terms of tile distributions, and thus as integer-valued equations.


    Given a pot P = {t_1, ⋯, t_n}, P realizes a graph G of order N if and only if there exists a tile distribution Ω(P) = (R_1, ⋯, R_n) such that
    

    ∑_j=1^nR_j = #V(G)   and   ∑_j=1^nR_jz_i,j = 0 for each i ∈{1, …, |Σ| }.
    
 













Finding the smallest graph that can be realized by a pot amounts to minimizing ∑_i=1^nR_i. Thus, we can see that the SRP is equivalent to the following integer programming problem.

    min   ∑_i=1^nR_i 
    subject to:    ∑_j=1^nR_jz_i,j = 0   ∀ i ∈_#P





Section <ref> details the computational tools we use to solve this integer program as well as example use cases. 



 §.§ Optimal Pot Problem Integer Program


Now we turn to solving the Optimal Pot Problem in Scenarios 1 and 2. The primary roadblock is the PRP (<ref>), i.e. being able to check whether a given pot realizes a given graph efficiently. This problem is known to be NP-hard, and since it will need to be answered for every possible pot of each size (of which there are combinatorially many), the number of necessary computations quickly becomes unreasonable. 

In this section, we reformulate the Optimal Pot Problem in Scenario 1 as an integer program. The overall approach will be to efficiently search the space of assembly designs of a given target graph G.

Recall that pot P realizes a graph if there exists an assembly design with assembling pot containing the tiles of P. Since we only care about minimal pots, we can restrict our attention to assembly designs whose assembly pots are equal to P. In other words, a pot P realizes a graph via assembly design λ if its realization function f (Definition <ref>) is surjective; i.e. f = P.

To solve the OPP, we will express the constraint f = P as a series of linear inequalities (see Definition <ref>). To do this, we must have some way of describing an assembly design λ as a fixed length integer vector. This vector will depend on the maximum possible size of a minimal pot realizing G, which we denote as θ, and the maximum possible number of bond-edge types in a minimal pot realizing G, which we denote as β. In an extension of the notation from Definition <ref>, we define Σ_β = {σ_1, σ_2, ⋯σ_β} and Σ̂_β = {σ̂_1, σ̂_2, ⋯σ̂_β}, so that Σ_β∪Σ̂_β will give all the half-edge labels we are considering.  


   Given an assembly design λ of G with realization function f such that f = P = {t_1, ⋯, t_N} for N ≤θ, Var(λ) is the following collection of decision variables describing λ:


    
  * t_n,a∈^+: the number of half-edges of type a on t_n
    
  * w_i,a∈^+: the number of half-edges of type a on xx v_i, or d_σ_i(v)
    
  * x_i,n∈{0, 1}: 1 if f(v_i) = t_n (v_i uses tile type t_n), and 0 otherwise 
    
  * k_n∈{0, 1}: 0 if t_n is an “empty" tile. This allows us to represent pots with less than θ tiles. 
    
  * e_i,j,a∈{0, 1}: 1 if  λ((v_i, {v_i, v_j})) = a (i.e. if the half-edge (v_i, {v_i, v_j}) is labeled with cohesive-end type a), 0 otherwise. 
    


Thus, to search over the space of labeled orientations of a given graph G, our integer program will search over the space of tuples: 
    C = {(t_n,a, w_i,a, x_i,n, k_n, e_i,j,a): a∈Σ_β∪Σ_β, i,j ∈_N, n ∈_θ}.

Each element of C can be encoded as an integer-valued vector with non-negative entries.


    Consider the realization of L=O(C_4) using the pot P = { t_1={a^2}, t_2={â^2}} pictured in Figure <ref>. The k_i and t_i,σ decision variables and values are shown on the left for each tile in the pot. Select e_i,j,σ decision variables are shown on the right, although most are omitted as every edge has 2 sets of variables for each σ∈Σ∪Σ̂. In the center, the x_i,j decision variables are shown for i=1 and i=3, and arrows demonstrate how these decision variables control the mapping of tiles onto vertices.
    
    


    
The following definition describes several linear constraints encoding the PRP in terms of the decision variables defined above.

 
    The following will be referred to as the Pot Construction Constraints on β bond-edge types, denoted PCC(β). The constraints describe pots realizing a graph G with at most β bond-edge types. Here, (X) → Y denotes that the constraint Y must be satisfied only if the Boolean expression X is true. 

    
  * 
    The net number of cohesive-end types is zero 
    
    ∑_i=1^Nw_i,σ-w_i,σ̂=0    ∀σ∈Σ_β

    
  * 
    The total number of cohesive-end types on a vertex (i.e. its number of half-edges) is equal to the degree of the vertex:
    
    ∑_σ∈Σ_β∪Σ_βw_i,σ=d_i     ∀ i∈ℤ_N

    
  * Each vertex is assigned exactly one tile type:
    
    ∑_n=1^Nx_i,n=1    ∀ i∈ℤ_N

    
  * Tile selections are enforced:
    
    (x_i,n=1)→ w_i,σ-t_n,σ≤ 0    ∀ i,n∈ℤ_N, σ∈Σ_β∪Σ_β

    
  * Each tile is assigned to a vertex of degree equal to the number of half-edges on the tile (indicator):
    
    (x_i,n=1)→∑_σ∈Σ_β∪Σ_βw_i,σ-t_n,σ=0     ∀ i,n∈ℤ_N

     
  * k_n decision variables are non-zero if and only if tile n is used (recall Δ(G) is the maximum degree of a vertex of G):
    
    -Δ(G) k_n+∑_σ∈Σ_β∪Σ_βt_n,σ≤ 0    ∀ n∈ℤ_N

    This constraint effectively enforces the following 2 conditions:
    
    1. If tile t_n is empty, k_n must be 0, because the total number of cohesive ends on tile n  is 0.
    
    2. If tile n is non-empty, the equation ensures that we get less than 0 on the left-hand side of the inequality for every vertex since the sum cannot be greater than Δ.  
    
  * Edge decision variables count the number of incident half-edges of each cohesive end-type:
    
    w_i,σ-∑_v_j∈ N_G(v_i)e_i,j,σ=0    ∀ i∈ℤ_N, σ∈Σ_β∪Σ_β

    
  * 
    Each edge is labeled with exactly two cohesive-end types:
    
    ∑_σ∈Σ_β∪Σ_βe_i,j,σ+e_j,i,σ=2    ∀ (v_i,v_j)∈ E(O(G))

    
  * 
    Each edge consists of two half-edges labeled with a hatted and unhatted version of the same bond-edge type:
    
    e_i,j,σ-e_j,i,σ̂=0     ∀ (v_i,v_j)∈ E(G), σ∈Σ_β∪Σ_β





    The constraints are written to allow integer programming solvers as much flexibility as possible in finding the optimal solution, and as such may be less specific than necessary.
 

The following theorem proves that the PCC constraints are equivalent to the constraint that for a given pot P and a given realization function f, f = P, or in other words, that P realizes G via f. 


    A pot P = {t_1, ⋯ t_N} realizes a graph G using β bond-edge types if and only Var(λ) satisfies PCC(β).

 


    Assume that a pot P = {t_1, ⋯ t_n} realizes a graph G with realization function f. We prove that Var(λ) satisfies all of the pot construction constraints given in Definition <ref>. 
     
     C1 follows from Lemma <ref>. C2 follows from the definition of labelled degree. C3, C4, C5, C7, C8, and C9 directly follow from the definition of a realization function (the fact that it is well defined and surjective on P).
     
     For C6, we must consider two cases that depend on the value of k_n. If k_n = 0, then t_n is an empty tile, and Σ_σ∈Σ_β∪Σ̂_β t_n,σ = 0. If k_n = 1, then there exists a vertex v_i such that f(v_i)=t_n, which implies x_i,n=1. From C2 and C5 we obtain the following.
     
    ∑_σ∈Σ_β∪Σ_β t_n,σ = ∑_σ∈Σ_β∪Σ_βw_i,σ = d_i

     Since Δ(G) ≥ d_i,
     
    -Δ(G) k_n+∑_σ∈Σ_β∪Σ_βt_n,σ  = -Δ(G) + d_i ≤ 0.
 In either case, we have shown the inequality holds.

     Conversely, now assume that the decision variable tuples 
    C = {(t_n,σ, w_i,σ, x_i,n, k_n, e_i,j,σ ) |σ∈Σ_β∪Σ_β, i,j,n ∈_N}
 satisfies PCC(β). 
     
     We construct assembly design λ so that Var(λ) = C using the e_i, j, σ variables.
     
     Let λ((v_i, {v_i,v_j}))=a if and only if e_i,j,a=1.

    First, we must show that λ is well defined. Thus, we must show λ((v_i, {v_i,v_j})) is the corresponding hatted symbol to λ((v_j, {v_i,v_j}))  (see Definition <ref>). For example, if λ((v_i, {v_i,v_j})) = σ_1, then λ((v_j, {v_i,v_j})) = σ̂_̂1̂
    Rearranging the sum in order to group the hatted and unhatted terms together in C8 gives the following. 
    
    ∑_σ∈Σ_β∪Σ_βe_i,j,σ + e_j,i,σ = ∑_σ∈Σ_β∪Σ_βe_i,j,σ + e_j,i,σ̂ = 2

   C9 ensures that e_i,j,σ = e_j,i,σ̂, so exactly one of the terms e_i,j,σ + e_j,i,σ̂ is equal to 2, which shows that both e_i,j,σ and e_j,i,σ̂ are 1, so  λ is well defined.
   
   Now, we show that Var(λ) = C. The e_i,j,a decision variables in C and Var(λ) are the same by construction. C7 ensures that w_i, σ equals the labelled degree of v_i, so the w_i,a decision variables are the same in C and Var(λ). C4, C5, and C6 ensure that the t_n,a, k_n, and x_i,n decision variables match, as well.
    
   C4 also tells us that for each i, there is some unique n_i such that x_i, n_i = 1. Combining this with C5, we get that t_n_i,σ = w_i,σ. Since f(v_i) is a tile with cohesive ends w_i, σ, f(v_i) = t_n_i as desired.




 §.§ Scenario 1 OPP Algorithm


In Scenario 1, we can restrict ourselves to β = 1, since all optimal pots in Scenario 1 use only one bond-edge type <cit.>. Thus, we employ the following integer program to solve the OPP. 


    min_C ∑_n=1^|G|k_n  subject to PCC(1)

where C ranges over all tuples 
    {(t_n,b, w_i,σ, x_i,n, k_n, e_i,j,b): λ∈Σ_1∪Σ_1, i,j,n ∈_|G|}
. This gives rise to the following algorithm.



   Algorithm <ref> solves the OPP in Scenario 1.


    A solution to the Scenario 1 integer program satisfies PCC(1), so by Theorem <ref> there exists an assembly design λ such that Var(λ) = C and the pot with tiles t_n, where t_n has t_n,σ cohesive end types of type σ, realizes G. 
    
    
    
   Now, suppose there is a pot P' realizing G such that #P' < #P. Let λ' be an assembly design corresponding to P'. Then, by Theorem <ref>, Var(λ') satisfies PCC(1). The sum of the k_n decision variables in Var(λ'), which counts the number of tile types of P', would be less than the sum of the k_n decision variables in Var(λ), thus contradicting the minimality of C.




 §.§ Scenario 2 OPP Algorithm


Now we consider the Optimal Pot Problem in Scenario 2. An ideal approach to Scenario 2 would combine a pot generation integer program similar to Algorithm <ref> with the SRP integer program given in (<ref>) into one integer program. Unfortunately, since the two integer programs employ different decision variables, this would require the ability to multiply decision variables, making the problem non-convex. Additionally, the objective functions for the two integer programs are conflicting, which creates a bilevel optimization problem. These two challenges make solving a single Scenario 2 integer program computationally infeasible. Instead, we will not combine Scenario 2 into a single integer program - we will run the integer programs separately. 

In the algorithm presented here, a pot generation integer program similar to Algorithm <ref> iteratively generates possible realizations of our graph, and then the SRP integer program given in (<ref>) determines whether each generated realization satisfies the constraints of Scenario 2.  If the SRP integer program determines that a pot realizes a graph of smaller order than the target (and hence is not a valid Scenario 2 solution), then the constraints of the pot generation integer program are modified to ensure that the offending pot is not generated again. An overview of this process is shown in Figure <ref>.



In order to efficiently track which pots have already been generated, we seek to create an ordering on the collection of pots. Since pots are defined by their tiles, which are defined by their cohesive-end types, we can fully specify a pot with an integer vector whose entries correspond to the cohesive-end types of its tiles. First, we encode tiles as positional integer vectors. 

    The tile vector v⃗(t_n) of tile t_n in collection Var(λ) is the vector whose i-th component, where i is odd, is the number of cohesive-ends of the type σ_i, and whose j-th component, where j is even, is the number of cohesive-ends of type σ̂_j-1.
    
    v⃗(t_n) = ⟨ t_n,σ_1, t_n,σ̂_1, t_n,σ_2, t_n,σ̂_2, ⋯ t_n,σ_β, t_n,σ̂_β⟩.



To these vectors we can assign an integer in a straightforward manner.

     Let the q-value of t_n be tile vector v⃗(t_n) written as an integer in base Δ(G), or formally 
    ∑_i=1^|v|v(t_n)[i] * Δ(G)^|v| - i



A candidate for encoding for a pot P is the concatenation of all v⃗(t_n) for all t_n ∈ P. But, given that a pot is a set, we should expect all permutations of the tiles to yield the same pot vector. To make the pot vector invariant to permutations, we order the tile vectors lexicographically.


    Given any two integer vectors v⃗ = ⟨ v_1, ⋯ v_n ⟩ and s⃗ =⟨ s_1, ⋯ s_n ⟩, v <_lex s if there is an integer m such that 
    
        
  * 0 ≤ m < n
        
  * v_i = s_i for all i < m
        
  * v_m < s_m
     


Note that v⃗ <_lexs⃗ if and only if q(v) < q(s). 


     Let ⟨ v_1 |⋯| v_j ⟩ denote the concatenation of v_1, ⋯ v_j. Then the pot vector V(P) of P = {t_1, ⋯, t_n} is the concatenation of its tile vectors 
    
    V(P) = ⟨v⃗(t_j_1) | ⋯ | v⃗(t_j_n) ⟩
 
    where j_1, j_2, ⋯ j_n is the unique ordering of the tiles in decreasing lexicographic order (i.e. such that v⃗(t_j_i) <_lexv⃗(t_j_i-1)  for all 2 ≤ i ≤ n).


Now that we have a well-defined encoding of a pot as a vector, we can encode that pot uniquely as an integer. 

     The characteristic value of P, q(P), is V(P) written as an integer in base Δ(G). That is, 
    q(P) = ∑_i=1^#VV(P)[i] * Δ(G)^# V - i.
  



   The pot vector for the pot P = {aĉ, âb̂, bcĉ, b̂c^2} is as follows.
      
    V(P) = ⟨ 100001 | 010100 | 001011 | 000120 ⟩


    The characteristic value of P in base 10 is the following integer. 
    q(P) = 100,001,010,100,001,011,000,120
 


We will now use the pot vector to build constraints for a modified pot generation integer program that returns a valid pot with the smallest characteristic value. We can only compare characteristic values for fixed-size vectors, so we design our integer program to search pots with fixed numbers of bond-edge and tile types. To achieve this, we define the Pot Restriction Constraints.


     The following will be referred to as the Pot Restriction Constraints on θ tile types and β bond-edge types, denoted PRC(θ, β).


    
  * P has exactly θ tiles:
    
    t_θ + 1, σ = t_θ + 2, σ = ⋯ t_N, σ = 0     ∀σ∈Σ_β∪Σ̂_̂β̂

    
  * Each bond-edge type is used:
    
    ∑_n=1^θt_n,σ≥ 1     ∀σ∈Σ_β

    
  * Each tile type is used:
    
    ∑_i=1^Nx_i,n≥ 1     ∀ n∈ℤ_θ

    
  * The collection {t_n, σ} is such that the tiles are in correct lexicographic order. That is,
    
    q(t_n) - q(t_n+1) ≥ 1     ∀ n∈ℤ_N-1



Extending Theorem <ref> using these constraints is then fairly straightforward. 

 
    There is an assembly design λ so that vectors 
    t⃗_⃗i⃗ = ⟨ t_i, σ_1, t_i, σ̂_1, ⋯ t_i, σ_n, t_i, σ̂_n⟩
 assemble into a valid pot vector ⟨t⃗_⃗1⃗|⋯|t⃗_⃗n⃗⟩ with β bond-edge types and θ tiles if and only if Var(λ) satisfies PCC(θ) and PRC(θ, β).


We are now ready to define the Scenario 2 pot generation integer program. The goal of this program is to find a pot with the minimum characteristic value that realizes G with β bond-edge types and θ tile types, such that the characteristic value q(P) is greater than some prespecified bound Q:

    min_C q(P) 
        subject to PCC(β), PRC(θ, β), q(P) ≥ Q


The algorithm provided below incorporates LP (<ref>) to find a valid Scenario 2 pot given bounds on the number of bond-edge and tile types.


We will now prove the correctness of Algorithm <ref>. We will need the following lemma.


    
    Let S = {P'_1, ⋯ P'_n} be the set of all pots P' such that:
    
        
  * P'_i realizes G with θ tile types and β bond-edge types.
        
  * P'_i is not a valid Scenario 2 pot, i.e. it realizes at least one smaller order graph.
        
  * q(P'_i) < q(P'_i+1) for all i such that 0 < i ≤ n 
        
  * q(P'_n) < q(P) for all i and any valid Scenario 2 pot P.
     
    Then P_i = P' and the algorithm continues to iteration i + 1 for all 1 ≤ i ≤ n



    Let P_i, Q_i, M_i be the values of P, Q, M in the ith iteration of Algorithm <ref>. 
    
    Now, since P'_1 has the smallest characteristic value among all non-valid Scenario 2 pots realizing G and by condition 4, has a smaller characteristic value than any valid Scenario 2 pot realizing G, P_1 is the solution to Integer Program <ref>, hence P_1 = P'_1. Since it is not a valid Scenario 2 pot, the algorithm does not terminate.

    Assume then that P_j = P'_j for j < i. Thus, at the ith iteration of the algorithm, Q_i = q(P_i-1) + 1 = q(P'_i-1) + 1
    
    Consider any pot P satisfying LP <ref> and not equal to P'_j for j < i. If P is not a valid S2 pot, then  q(P'_i) < q(P) since otherwise P = P'_j for j < i. If it is a valid Scenario 2 pot, condition 4 guarantees that q(P'_i) < q(P). Furthermore, q(P'_i) > q(P'_i-1) by condition 3. Thus, P'_i has minimal characteristic value among all pots realizing G with characteristic value greater than or equal to Q_i = q(P'_i-1) + 1, hence P_i = P'_i. By induction, the theorem is proved.



    Algorithm <ref> determines if there is a pot P with β bond-edge types and θ tile types such that G ∈𝒪(P) and for all H ∈𝒪(P), #V(H) ≥#V(G). If at least one such pot exists, Algorithm <ref> returns the pot with the minimum characteristic value Q^*.


    Let C_i, P_i, Q_i, M_i be the values of C, P, Q, M in the ith iteration of Algorithm <ref>. 

    First, assume that there is no valid Scenario 2 pot. If there is no pot satisfying LP (<ref>), then, Lemma <ref> implies that C_1 is empty and we will conclude that there is no valid Scenario 2 pot. If there are pots satisfying LP (<ref>), but all of those pots realize at least one graph of smaller order than G, then M_i < N in all iterations i of the algorithm. Since we increment Q at each iteration and the maximum characteristic value of all possible pots realizing G is finite, we will eventually create an instance of LP (<ref>) which has no solution, and hence the algorithm will also determine that there is no valid Scenario 2 pot.

    Conversely, assume that there is a valid Scenario 2 pot. Let S = {P'_1, ⋯, P'_n} be the set of all pots realizing G with a characteristic value less than Q^* ordered by characteristic value. S satisfies all of the conditions of Lemma <ref>, and so there will be an iteration P_n = P'_n, the element with the maximal characteristic value in S. Since P'_n∈ S, M_i < N, and hence the algorithm will continue to iteration i+1, where Q_i+1 > q(P') for any P' ∈ S by the maximality of P'_n. Therefore, P_i+1 is not in S, which means that it must be a valid Scenario 2 pot realizing G. The SRP integer program will verify that M_i+1 = N, so the algorithm returns a pot P_i+1, which is clearly the valid Scenario 2 pot with the minimal characteristic value.


We now have an algorithm that determines if there is a valid Scenario 2 pot with β bond-edge types and θ tile types. The final step to develop a full Scenario 2 algorithm is to search the space of possible tuples (β, θ) for an optimal pot. We rely on the following Theorem from <cit.> to increase search efficiency.

    If G is a graph with n>2 vertices, B_2(G)+1≤ T_2(G) 


Theorem <ref> implies that it is unnecessary to search tuples (β, θ) where β > θ. It is sufficient to search the remaining region in order of increasing number of tile types (θ) to find a pot achieving (β, T_2(G)) (see Figure <ref>). It is equivalent to search that same region in order of increasing number of bond-edge types to find a pot achieving (B_2(G), θ). We combine these procedures in the following algorithm.








§ RESULTS AND DISCUSSION


We have implemented the algorithms described in Sections <ref>, <ref> and <ref> in Python using the Gurobi MIP solver, which was chosen for its fast runtime and ability to handle many indicator constraints <cit.>. The Oriented Optimal Pot Solver ( https://github.com/JacobAshw/OOPSOOPS) for solving the OPP and the Subgraph Realization Problem Solver (SRPS) for solving the SRP are both available along with command-line interfaces and visualization tools to help with ease of use.



 §.§ Implementation of SRP Algorithm

Even pushing the bounds beyond pots that would realistically be encountered (such as the 27-tile pot or the pot with a 200 half-edge time in Table <ref>), we see that our proposed algorithm is able to compute the minimal realization size of a graph in less than a tenth of a second. This is due to the fact that the SRP integer program is small, with a single constraint for each bond-edge type in the pot. Since integer programs are still relatively easy to solve with constraints numbering in the hundreds (see Table <ref> for selected runtimes for the Scenario 1 algorithm), this means the SRP algorithm has a very high ceiling for solvable pots and is not a limiting constraint on the runtime of our larger Scenario 2 algorithm.



 




 §.§ Implementation of Scenario 1 OPP


We use OOPS to compute optimal pots in Scenarios 1 and 2 for various graphs. This serves three separate purposes:

    
  * Algorithmic verification: We ensure the optimality of known pots.
    
  * Runtime analysis: Since the complexity of an integer program varies greatly depending upon the structure of the specific integer program, runtimes for example graphs give a general idea of the scope of graphs for which we can feasibly compute optimal pots.
    
  * Discovery of new optimal pots: We search for optimal pots for previously unsolved graphs.


In each scenario, we provide two tables. The first (Tables <ref> and <ref>) compares the algorithm's output on previously solved graphs to ensure that the output aligns with the expected results. The second (Tables <ref> and <ref>) displays computed optimal pots on graphs that were previously unsolved in the specified scenario. The specific graphs are chosen somewhat arbitrarily, but represent a range of possible structures for the purpose of demonstrating the algorithm's potential.

In Scenario 1,  we focus exclusively on optimizing tile types. Table <ref> shows our algorithm's computed optimal pots, which demonstrate agreement with the proven T_1 values of various other papers. Table <ref> shows the results of the algorithm on a set of various new graphs. These graphs include small examples of social networks, Cartesian products of two different graphs, and disconnected graphs. Each of these sets of graphs was chosen because they represent a possible new area of exploration in finding optimal pots, and they have varying structures that will challenge the algorithm in different ways.

Of the tested graphs, only Les Misérables, K_100, and Karate Club × Petersen have runtimes greater than one second. These are all graphs of substantial order, having 77, 100, and 170 vertices respectively. Furthermore, the Les Misérables graph is slower (×1.62) than the K_100 graph, despite having far fewer edges (242 versus 4950). This suggests that the runtime is generally dependent on the number of vertices rather than the number of edges. This can be explained by the fact that C4 and C5 of the pot construction constraints each enforce N^2 constraints, so the number of constraints grows quadratically with the number of vertices.

We have also used this algorithm to find optimal pots in Scenario 1 for disconnected graphs. Since Scenario 1 is not concerned with the construction of smaller graphs, the question of an optimal pot for a disconnected graph is well-posed. At the time of this writing, there are no known previous results for disconnected graphs in the flexible-tile model. The optimal pot is not always simply the union of the optimal pots for each component, as proven by the disconnected graph consisting of K_4, S_3, and P_2. It is known that the optimal pots of these three graphs in Scenario 1 are {a^2â,aâ^2},{a^3,â}, and {a,â}, respectively, which under a union create a pot P with #Σ (P) = 4 (as shown in Figure <ref>). Our algorithm finds a pot P realizing this graph with P with #Σ (P) = 3 (as shown in Figure <ref>). This opens the door for exploration of ways in which optimal pots for disconnected graphs relate to the optimal pots of graph's components. 











 §.§ Implementation of Scenario 2 OPP

In Scenario 2 there are two different versions of optimality: tile optimality and bond-edge optimality. Using the search process described in Section <ref>, our algorithm first optimizes over tile types, and then verifies the computed value for bond-edge types. In general, we see that the runtime of the algorithm grows with the size of the solution (so values of T_2(G) and B_2(G)) rather than the size or order of the graph in question. For instance, the path graph P_10 has a much longer runtime than the complete graph K_100 despite having a far smaller size (9 vs. 4950) and order (10 vs 100) (as shown in Table <ref>). This is because the optimal pot for the path graph consists of six tile types and five bond-edge types, whereas the optimal pot for K_100 consists of only two tile types and 1 bond-edge type.

In addition to reproducing many known optimal pots in Scenario 2 (Table <ref>), we use our algorithm to compute many novel optimal pots. One solution of notable interest is the optimal pot for the dodecahedron, as it was the only remaining unsolved platonic solid in Scenario 2 <cit.>. The realization of this graph using the optimal pot can be seen in Fig. <ref>. Other examples of novel solutions we computed are listed in (Table <ref>). We provide optimal pots but do not provide realizations of each graph here (for the purposes of saving space), although all results can be easily reproduced by downloading and running https://github.com/JacobAshw/OOPSOOPS (found at https://github.com/JacobAshw/OOPS), which displays a visual representation of the graph realization upon computation of the optimal pot. 

It is important to note that the bond verification step of the program only improves upon the solution computed in the tile runtime step when no biminimal pot exists, i.e. the optimal number of tile and bond-edge types cannot be achieved in the same pot. In Scenario 2, an example of a graph that exhibits this property is Y_3,3 (the stacked prism graph), discovered by Toby Anderson, Olivia Greinke, Iskandar Nazhar, and Luis Santori (pending citation). In practice this property is exceedingly rare (currently Y_3,3 is the only known example), so the bond verification step nearly never improves upon the pot found in the first step, despite taking much longer.










§ CONCLUSION


This paper describes three separate computational tools to tackle some of the combinatorial problems associated with DNA self-assembly. The first tool, the SRP integer program, solves the Subgraph Realization Problem for any given pot. The second tool, the Scenario 1 integer program, solves the OPP in Scenario 1 on large graphs. The third tool, the Scenario 2 integer program, solves the OPP in Scenario 2, although it is much slower than the Scenario 1 integer program, and therefore cannot handle graphs with complex optimal pots. These tools provide researchers working on theoretical flexible tile self-assembly with ways to generate and verify solutions that were not previously possible. The SRP integer program allows researchers to verify the optimality of pots in Scenario 2, and the Scenario 1 and 2 integer programs allow for the generation of optimal pots which researchers may be able to generalize to wider families of graphs.

An immediate natural extension of this work is the creation of a Scenario 3 integer program. To do this, two primary challenges must be addressed. First, such a program would require the utilization of a computational tool to solve the non-isomorphic graph realization problem which defines Scenario 3. Once such a tool exists, it can be used similarly to the Scenario 2 integer program to create an additional verification loop in the algorithm. The second challenge is the growing search space. In general, Scenario 3 pots tend to be larger than those in Scenario 2. Since the runtime of our algorithm depends on the complexity of the optimal pot, this will result in significant slowdown. 

There is also opportunity for future work in optimizing the Scenario 2 integer program. Several inefficiencies remain apparent in the algorithm presented here. Many of the different pots generated by the Scenario 2 integer program are equivalent (up to swapping tile labels), which results in the unnecessary checking of a significant number of pots. Additionally, the Scenario 2 integer program is still separated into two scripts. Unifying these into a single script could greatly improve efficiency.

Finally, it is possible to use the algorithms presented here to fully explore optimal solutions across broader families of graphs. Specifically, this could be used to algorithmically generate graphs for which a biminimal pot does not exist in Scenario 2. While it is known that such examples exist, there are no known characterizations of graphs with this property. This is also of great interest to the development of the algorithm itself, as the slowest step of the algorithm is the verification of B_2(G). This step would become unnecessary in the vast majority of cases if it were possible to accurately predict which graphs will exhibit this property.



§ ACKNOWLEDGEMENTS


We would like to thank faculty and staff at ICERM. Specifically, we want to thank the  the Summer@ICERM TAs, Ryan Firestine, Michaela Fitzgerald, Eric Redmon, and Steph Reyes for their support. We would also like to thank our fellow students who participated in the Summer @ ICERM 2023 REU program. 




§ FUNDING SOURCES


This material is based upon work supported by the National Science Foundation under Grant No. DMS-1929284 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the Summer@ICERM program. 












































elsarticle-num ]

 Paper Title: Biased domination games.



Robust Implementation of Discrete-time Quantum Walks in Any Finite-dimensional Quantum System
    Amlan Chakrabarti^1
    August 5, 2024
=============================================================================================




§ ABSTRACT 
We consider a biased version of Maker-Breaker domination games, which were recently introduced by Gledel, Iršič, and Klavžar.
Two players, Dominator and Staller,
alternatingly claim vertices of a graph G
where Dominator is allowed to claim up to b
vertices in every round and she wins if and only if she occupies all vertices of a dominating set of G.
For this game, we prove a full characterization
of all trees on which Dominator has a winning strategy. For the number of rounds which Dominator needs to win, we give exact results when played on powers of paths or cycles, 
and for all trees we provide bounds which are optimal up to a constant factor not depending on b.
Furthermore, we discuss general minimum degree conditions and study how many vertices can still be dominated by Dominator even when Staller has a winning strategy.





§ INTRODUCTION


A dominating set D in a graph G=(V,E) is a set of vertices
such that each vertex in V∖ D has a neighbour in D.
The domination number γ(G) of a graph G, i.e. the smallest size of a dominating set of G, belongs to the most prominent graph parameters in graph theory. While the decision problem whether γ(G)≤ k holds is a classical example of an NP-complete decision problem <cit.>, research in the last decades has brought many interesting results, including e.g. concentration results for random graphs <cit.>.

Note that a greedy algorithm, such as taking large-degree vertices one at a time, can lead to dominating sets that are very far apart from the optimum. Nevertheless, to model such a vertex-by-vertex approach, Brešar, Klavžar, and Rall <cit.>
introduced the domination game played on a graph G as follows. Two players, called Dominator and Staller, alternatingly choose vertices of G with the restriction that any new vertex must enlarge the set of already dominated vertices, and the game ends once all vertices are dominated by the chosen vertices.
Dominator's goal is to minimize the number of moves,
while Staller's goal is to maximize this number.
The game domination number then is the smallest number of moves until the game ends, under optimal play of both players,
and it is denoted with γ_g(G) or γ_g'(G),
depending on whether Dominator or Staller is the first player. 

Since the seminal paper <cit.>,
the above combinatorial game has been studied extensively,
including challenging open problems such as the 3/5-conjecture
for graphs without isolated vertices <cit.> and the 1/2-conjecture for traceable graphs <cit.>.
Nevertheless, and partly due to the non-monotone behaviour of these games <cit.>,
only few precise results are known for fixed graphs G, including
paths and cycles <cit.>, powers of cycles <cit.>, and caterpillars <cit.>.
Unfortunately, even for such simple graph classes, the study of the domination game tends to be complex;
and already for the class of trees no precise results are known for the parameters γ_g(G) and γ_g'(G) in general. For an overview on game domination numbers and variants we recommend the book <cit.>.



Very recently, Gledel, Iršič, and Klavžar <cit.> introduced the following Maker-Breaker variant of the above game.
Two players, again called Dominator and Staller,
alternatingly claim vertices of the given graph G.
Dominator wins if she manages to claim all the vertices of a dominating set of G, and Staller wins otherwise.
One main question then is to characterize all graphs for which Dominator has a winning strategy.
Provided that Dominator can win,
the other main question then is how fast she can do so. 
For this, let the Maker-Breaker domination number 
γ_MB(G) be the smallest number of rounds within which Dominator can always win the Maker-Breaker domination game on G, provided that she is the first player and both players play optimally, and
set γ_MB(G)=∞ if Staller has a winning strategy. Moreover, define γ'_MB(G) analogously for the case when Staller is the first player.

Although the rules of the above two types of games seem very similar, they can behave very differently.
For instance, it is known that |γ_g(G)-γ'_g(G)|≤ 1
for every graph G <cit.>,
while γ_MB(G) and γ'_MB(G) can be arbitrarily
far apart <cit.>.



For any integers 2≤ r≤ s≤ t, there exists a graph G such that γ(G)=r, γ_MB(G)=s
and γ_MB'(G)=t.


Moreover and amongst others, there exists a full characterization 
of all trees on which Dominators wins the Maker-Breaker domination game <cit.>, and precise results
on γ_MB and γ_MB' were proven
for all trees and cycles <cit.>.
To describe their result,
Gledel et al. define a residual graph R(G) which is obtained from G by iteratively removing pendant paths of length 2 until 
reaching a graph which does not have any further such pendant paths.



If T is a tree, then

    γ_MB(T) =
    v(T)/2,     if T has a perfect matching
    v(T)-1/2,     if R(T) is a single vertex
    v(T)-k+1/2,     if R(T) is a star with k≥ 3 edges
    ∞,     otherwise

and

    γ_MB'(T) =
    v(T)/2,     if T has a perfect matching
    ∞,     otherwise.




For every integer n≥ 3,

    γ_MB(C_n) =
    γ_MB'(C_n) =
    ⌊n/2⌋ .





Motivated by the literature on Maker-Breaker games,
it seems natural to extend the above games by introducing a bias,
and with this paper we intend to prove first results in the case when a bias is given to Dominator. 
We note that we are not aware of any
papers considering domination games with a bias,
except from <cit.>
which considers a variant called
Disjoint Domination Game.



In general, a  biased (p:q) Maker-Breaker game on a hypergraph
(X,ℱ) is defined as follows:
Maker and Breaker alternatingly claim vertices of the board X, with Maker always claiming up to p vertices before Breaker
claims up to q vertices. Maker wins if she occupies
a winning set from ℱ completely,
and Breaker wins if he can prevent such a situation throughout the game. For an overview on such games we recommend
the monograph <cit.> and the survey <cit.>.

In these terms, we can define the b-biased Maker-Breaker domination game on a graph G as the (b:1) Maker-Breaker game
where X is the vertex set of G and
ℱ contains all dominating sets of G.
To be consistent with the previously introduced game, we call Maker the Dominator and we call Breaker the Staller.
Moreover, we define 
γ_MB(G,b) and γ'_MB(G,b)
as the smallest number of rounds in which Dominator
wins the b-biased Maker-Breaker domination game on G
when Dominator or Staller start, respectively,
and where we set such a value to be ∞ if Staller
has a winning strategy.

As our first result, we extend the known results on paths and cycles by considering b-biased Maker-Breaker domination games on powers of paths and cycles. For this note that the k-th power G^k
of a graph G is obtained by putting an edge between any two vertices of distance at most k.


For all integers b,k ≤ n it holds that

    γ_MB(P_n^k,b) = 
    ⌈n-1/b(2k+1)-1⌉  and  γ_MB(C_n^k,b) = 
    ⌈n-1/b(2k+1)-1⌉ .



Additionally, we give a full characterization of all trees
on which Dominator wins the b-biased Maker-Breaker domination game. Given a tree T and a bias b, we say that T is a b-good tree
if we can recursively delete vertices, which have exactly b leaf neighbours, and also delete these leaf neighbours,
until we reach a forest where every vertex has at most b-1 leaf neighbours. Then the following holds.


Let T be a tree with v(T)≥ 2, and let b≥ 1 be any integer. Then the following are equivalent:


  (i) Dominator wins the b-biased Maker-Breaker domination game on T when Staller is the first player.

  (ii) T is b-good.



While the implication (i)⇒ (ii) is a simple exercise,
the proof of the opposite direction will be done by a more involved induction of a more general statement. Details will be given later in Section <ref>. Moreover, the more general statement in this section will also allow for an analogous characterization for the case when Dominator is the first player.

Additionally, for the trees in which Dominator cannot win, we prove the following quantitative statement.


For every tree T it holds that Dominator can dominate
at least (1-1/(b+1)^2) v(T) vertices in the 
b-biased Maker-Breaker domination game on T. Moreover, the bound is sharp.


Looking at Theorem <ref>, we see that,
when Staller starts, γ'_MB(T)
is of the same size for every tree T on n vertices for which γ_MB'(T)<∞ holds. In the biased case, this changes drastically. Specifically, we prove the following.


Let b≥ 2 be any integer. Then the following holds.


  (a) For every tree T on n vertices, if γ_MB'(T,b)<∞ then

    ⌈n/b(b+3)⌉≤γ_MB'(T,b) ≤⌈n/b+1⌉.


  (b) 
Let f(b):=(⌊b/2⌋ + 1)(⌈b/2⌉ + 1). Then
for every integer t with 
⌈n/f(b)⌉≤ t ≤⌈n/b+1⌉ there exists a tree T on n vertices such that
γ_MB'(T,b) = t.



Note that the lower bound in part (a) is optimal up to a constant factor 4, due to the trees guaranteed by part (b) and since
4f(b)>b(b+3) for every b.
Moreover, a short discussion on γ_MB(T,b)
will be given later in this paper.



Finally, we also aim to provide some general statements on
b-biased Maker-Breaker domination games. The next two results in this regard give a tight minimum degree condition for Dominator to have a winning strategy and a general upper bound on γ_MB(G,b).


Let n be a positive integer.
If G is a graph with minimum degree δ(G) > log_b+1(n) - 1,
then Dominator wins the b-biased Maker-Breaker domination game on G.

Moreover, this bound on δ(G) is asymptotically best possible.
For infinitely many n, there is a graph G on n vertices
and with δ(G) > log_b+1(n) - 2 such that Staller wins
the b-biased Maker-Breaker domination game on G.



Let G be a graph on n≥ 2 vertices  with 
minimum degree δ(G)≥ 10ln(n),
and let b≥ 1 be an integer. Then

    γ_MB(G,b) ≤10nln(n)/bδ(G)  .



Note that the above bound on γ_MB(G,b) is optimal up to a logarithmic factor. This can be easily seen by considering a graph G on n vertices which is the disjoint union of copies of K_δ+1, since then δ(G)=δ and γ_MB(G,b)=⌈n/b(δ + 1)⌉. 




Organization of the paper. 
At first, in Section <ref>,
we recall Beck's winning criterion for Maker-Breaker games.
Then, in Section <ref>, we prove Theorem <ref> on powers of paths and cycles. In Section <ref> we discuss Maker-Breaker domination games for trees,
including the proofs of Theorem <ref>,
Theorem <ref>
and Theorem <ref>.
Afterwards, in Section <ref>
we prove Theorem <ref>
and Theorem <ref>
providing minimum degree conditions for biased Maker-Breaker domination games.
We finish the paper with some concluding remarks in
Section <ref>.



Notation. 
Most of the notation used in this paper is standard 
and is chosen according to <cit.>.
We set [n]:={k∈ℕ:  1≤ k≤ n} for
every positive integer n. 
Let G be a graph. Then we write V(G) and E(G) for the vertex  set and the edge set of G, respectively, and we set 
v(G) :=|V(G)| and e(G):=|E(G)|.
Additionally, if we have a family
Q=(G_1,…,G_k) of graphs we
let V(Q)=⋃_i∈ [k] V(G_i)
and v(Q):=|V(Q)|.
Let v,w be vertices in G.
If {v,w} is an edge in G,  
we write vw for short. 
The distance between v and w
is the smallest number of edges
among all paths between v and w in G.
The neighbourhood of v is
N_G(v) : ={w∈ V(G): vw∈ E(G)},
and we call its size the degree of v.
We write
N_G[v]:=N_G(v)∪{v}
for the closed neighbourhood of v.
Moreover, the minimum degree in G is denoted δ(G).
Given any A,B⊂ V(G),
we set
N_G(A):= (⋃_v∈ A N_G(v))∖ A, 
N_G[A]:= N_G(A) ∪ A, and
e_G(A,B):=|{vw∈ E(G):  v∈ A,w∈ B}|.
Often, when the graph G is clear from the context,
we omit the subscript G in all definitions above.
 
Let G and H be graphs. 
We say that H is a subgraph of G,
denoted H⊂ G, if both V(H)⊂ V(G) and 
E(H)⊂ E(G) hold.
For a subset A⊂ V(G) we let G-A
be the subgraph of G 
with vertex set V(G)∖ A
and edge set {vw∈ E(G):  v,w∈ V(G)∖ A}.
Moreover, if v∈ V(G), then 
G-v:=G-{v}.

Let F be a forest. Then we write L(F) for the set of leaves, i.e. all vertices of degree 1 in F. 
We write P_n for the path on n vertices, i.e. with vertex set V(P_n)={v_1,…,v_n}
and edge set E(P_n)={v_iv_i+1:  i∈ [n-1]}.
Similarly,
we write C_n for the cycle on n vertices, i.e. with vertex set V(C_n)={v_1,…,v_n}
and edge set E(C_n)={v_iv_i+1:  i∈ [n-1]}∪{v_nv_1}.

We write Bin(n,p) for the binomial random variable
with n trials, each having success independently 
with probability p. We write X∼Bin(n,p) to   
denote that X is distributed like Bin(n,p),
and we say that an event, depending on n, holds asymptotically almost surely (a.a.s.) if its probability
tends to 1 as n tends to infinity.
Moreover, for functions f,g:ℕ→ℝ, we write f(n)=o(g(n)) if
lim_n→∞ |f(n)/g(n)| = 0.

Assume a Maker-Breaker domination game is in progress, then we call a vertex free if it is not claimed by any player so far.










§ PRELIMINARIES


Later we will use the following version of a result due to Beck <cit.>, stated in the way
that (i) Breaker is the first player 
and additionally (ii) Breaker wants to bound the number of winning sets that Maker can occupy.
The proof of this version is almost the same as in <cit.>.
In <cit.> (see Theorem 3.2.1 therein) and in <cit.> (see comment before Lemma 5 therein) it can be found 
how Beck's original argument can be adapted to
allow for one of the two options (i) and (ii).
Moreover, it is straightforward to combine both options. 


Let q∈ℕ, and let (X,) be a hypergraph. Then playing a (1:q) Maker-Breaker game
on (X,), Breaker can ensure that
Maker occupies at most

    ∑_F∈ℱ (1+q)^-|F|

winning sets, provided he is the first player.


Moreover, for some probabilistic arguments, we will use the following
Chernoff bounds (see e.g. <cit.>).


	If X ∼Bin(n,p), then
	
    		
  * Prob(X<(1-δ)np)
    				< exp(-δ^2np/2) 
    			for every δ>0, and
    		
  * Prob(X>(1+δ)np)< exp(-np/3) 
    			for every δ≥ 1.
	













§ POWERS OF PATHS AND CYCLES


In this section we prove Theorem <ref>.
Throughout the proof, we set N:=b(2k+1)-1.



Upper bound for P_n^k.
Let s∈ℕ be such that 2≤ s≤ N+1 and s ≡ n (mod N). Let v_1,v_2,…,v_n be the vertices
of P_n.
In the first round, Dominator claims the vertices 
v_k+1+(2k+1)i with 0≤ i≤ b-1, and in particular
dominates the first s vertices v_1,…,v_s.
Afterwards, consider the graph induced by
v_s+1,…,v_n, and note that its number of vertices is a multiple of N. From now on we can play only on this graph, assuming that Dominator is
the second player and no vertex has been claimed so far.
Split the vertex set into intervals of length N,
i.e. I_i = {v_s+1+(i-1)N,…,v_s+iN} with i∈n-s/N.
In each of the next rounds r=1,…,n-s/N, Dominator always chooses a distinct interval
and dominates it completely. In case that Staller touched
an interval for the first time in the previous round while this interval is not yet dominated by Dominator, Dominator chooses the same interval.

So, consider any such interval I_i of length N, and assume that Staller claims
(at most) one vertex v in it. It is easy to see that
the interval can be split into b subintervals such that
all but one subinterval have length 2k+1,
and the vertex v is either in the unique subinterval of length 2k,
or in one of the other subintervals but not in its middle.
Having such a partition, Dominator can always find a free element in each
of the subintervals with which the whole subinterval can be dominated.
Hence, by claiming only b vertices, Dominator can dominate the whole interval I_i.

The number of rounds played is
1+n-s/N = ⌈n-1/N⌉.
Hence, γ_MB(P_n^k,b)≤⌈n-1/b(2k+1)-1⌉. 



Lower bound for P_n^k.  
For the desired lower bound, we will apply an inductive argument,
but for a slightly stronger statement.
Let 𝒬=(Q_1,…,Q_ℓ) and
ℛ=(R_1,…,R_ℓ) be families of 
vertex-disjoint paths. We write 𝒬⊂ℛ 
if Q_i⊂ R_i for every i∈ [ℓ].
In this case we define the game 
D_MB(𝒬,ℛ,k,b) 
as follows: Maker (first player) and Breaker claim vertices of ℛ according to bias (b:1), and Maker wins if and only if she occupies a vertex set F⊂ V(ℛ) such that each vertex of V(𝒬) has distance at most k to the set F. That is, the family of winning sets is

    ℱ:= {F⊂ V(ℛ):  (∀ v∈ V(𝒬) ∃ w∈ F: dist(v,w)≤ k) }.

Moreover, we denote with  γ_MB(𝒬,ℛ,k,b) the smallest number of rounds in which Maker can win the game, and we change the subscripts to BM if Breaker is the first player.

Inductively we will prove the following more general statement.


For all families 𝒬=(Q_1,…,Q_ℓ) and
ℛ=(R_1,…,R_ℓ) of 
vertex-disjoint paths such that 𝒬⊂ℛ, 
and for all integers k,b we have

    γ_MB(𝒬,ℛ,k,b)
    ≥⌈v(𝒬)-1/b(2k+1)-1⌉   and   γ_BM(𝒬,ℛ,k,b)
    ≥⌈v(𝒬)/b(2k+1)-1⌉ .



Note that by setting 𝒬=ℛ={P_n} in the above claim, the desired bounds
for the Maker-Breaker domination game on P_n^k follow.


We apply induction on V(𝒬)
and note that the claim is true if v(𝒬)≤ N,
since γ_MB(𝒬,ℛ,k,b)≥ 1 and γ_BM(𝒬,ℛ,k,b)≥ 1 hold trivially.
For the induction step, we separately consider the cases
that Maker starts the game or Breaker is the first player.

Case 1: Maker starts.
In the first round Maker claims up to b vertices.
Afterwards, we can colour these vertices red,
and use colour blue for all other vertices which have distance at most k to a red vertex.
Then we let 𝒬'=(Q_1',…,Q_ℓ'') be a family consisting of all maximal paths induced by the yet uncoloured vertices, and note that
v(𝒬')≥ v(𝒬)-b(2k+1) = v(𝒬) - N - 1.
Moreover, we define the family 
ℛ'=(R_1',…,R_ℓ'') as follows: 
we extend each path Q_i' to a path R_i' by adding back all the blue vertices from V(ℛ) that are connected to Q_i even after all red vertices are deleted. Note that the blue vertices in a path R_i' have distance larger than k to every path Q_j' with j≠ i.
Hence, Breaker can consider to play the game D_BM(𝒬',ℛ',k,b) and play a strategy which
prevents that Maker wins within less than 
γ_BM(𝒬',ℛ',k,b) further rounds.
By induction, we thus obtain

    γ_MB(𝒬,ℛ,k,b) 
    ≥ 
    1 + γ_BM(𝒬',ℛ',k,b)
    ≥
    1 + ⌈v(𝒬')/N⌉≥⌈v(𝒬')+N/N⌉≥⌈v(𝒬) - 1/b(2k+1)-1⌉ .


Case 2: Breaker starts.
If all paths in 𝒬 have at most 2k vertices,
then the number of paths in 𝒬 is at least v(𝒬)/2k and thus
γ_MB(𝒬,ℛ,k,b) 
≥v(𝒬)/2bk≥v(𝒬)/N.
Hence, from now on we can assume that 𝒬
contains a path, say Q_1, of length at least 2k+1.
Label the vertices of Q_1 with v_1,v_2,…,v_s
such that s≥ 2k+1,
and label the additional vertices of R_1 with
…,v_-1,v_0 (vertices attached to v_1)
and v_s+1,v_s+2,… (vertices attached to v_s).

In the first round Breaker claims v_k+1.
Afterwards and as long as Maker does not claim
a vertex v_i with i≤ k, Breaker claims the vertices
v_k,v_k-1,v_k-2,… in this order.
If Maker never claims a vertex v_i with -k+1≤ i≤ k,
she looses since she will then not manage to occupy a vertex of distance at most k to the vertex v_1.
So assume that for some r, round r is the first round
in which Maker claims a vertex v_i with -k+1≤ i≤ k.
At that moment, Breaker has already claimed all vertices
v_k-r+2,…,v_k+1, and hence i≤ k-r+1.
So, for v_i there can be at most 2k-r+1 vertices in V(𝒬) of distance at most k to v_i.
In particular, it follows that after Maker's r^th move, 
since for every other Maker's vertex there can be at most 2k+1 vertices of distance at most k,
there are in total at most

    (2k+1-r) + (rb-1)(2k+1) = r(b(2k+1)-1)=rN

vertices in 𝒬 which have distance at most k to a Maker's vertex.

Analogously to Case 1, but now after round r,
we define red vertices (claimed by Maker) and blue vertices (distance at most k to a red vertex), and using these colours,
we define 𝒬' and ℛ' as before.
By the previous estimate, we have
v(𝒬')≥ v(𝒬)-rN, and r rounds were played already.
Now, Breaker is the next player and w.l.o.g. he can assume that
all his previously claimed elements are free again
so that he can consider to play the game
D_BM(𝒬',ℛ',k,b) from now on.
Applying induction analogously to Case 1,
we can conclude

    γ_MB(𝒬,ℛ,k,b) 
    ≥ 
    r + γ_BM(𝒬',ℛ',k,b)
    ≥
    r + ⌈v(𝒬')/N⌉
    =
    ⌈v(𝒬')+rN/N⌉≥⌈v(𝒬)/b(2k+1)-1⌉ .




Bounds for C_n^k. 
When n≤ b(2k+1), then Dominator can dominate all vertices
within the first round, and thus γ_MB(C_n^k,b)=1.
Moreover, for every n we have
P_n^k⊂ C_n^k, from which it immediately follows that
γ_MB(C_n^k)≤γ_MB(P_n^k) = ⌈n-1/b(2k+1)-1⌉.
Hence, it is enough to give a strategy for Staller that prevents
Maker from winning the (b:1) Maker-Breaker domination game on C_n^k within less than ⌈n-1/b(2k+1)-1⌉ rounds, when n>b(2k+1).

After Maker's first move, at most b(2k+1) are dominated.
As before, colour all claimed vertices red, and colour all other dominated vertices blue. Looking only at the underlying path P_n, the uncoloured vertices induce a family of paths 𝒬' with v(𝒬')≥ n-b(2k+1), 
and using the blue vertices attached to these paths, 
we obtain a family ℛ' with 𝒬'⊂ℛ'. Now, the game can be reduced to D_BM(𝒬',ℛ',k,b), and we already know from Claim <ref> that 
γ_BM(𝒬',ℛ',k,b)≥⌈n-b(2k+1)/b(2k+1)-1⌉.
Hence, Staller can play in such a way that Dominator needs at least ⌈n-b(2k+1)/b(2k+1)-1⌉
further rounds for winning on C_n^k, leading to a total of at least 1+⌈n-b(2k+1)/b(2k+1)-1⌉ = ⌈n-1/b(2k+1)-1⌉ rounds.

















§ TREES




 §.§ Characterization

As a first step we want to prove Theorem <ref>
by induction.
In the induction step we then have to deal with forests
from which some set of vertices A
is already claimed by Dominator, and in particular N[A] is dominated. We therefore extend the definition of b-goodness, as given immediately before Theorem <ref>, in such a way that it considers such a set A. Specifically, we will use the following notions. Recall that L(F) denotes the set of leaves of a given forest F.


Let T be a tree, let A⊂ V(T), let b∈ℕ, and  let 𝐯=(v_1,v_2,…,v_t) be a sequence
of distinct vertices in V(T)∖ A. 
Then we define a forest T_↓𝐯,A
as follows:
Set F_0:=T and  
F_i := F_i-1 - ({v_i}∪ (N(v_i)∩ L(F_i-1)∖ A))
for every i∈ [t].
Finally, let T_↓𝐯,A := F_t.
Moreover, we define the following:


  * We say 𝐯 is (A,b)-admissible for T if 
|N(v_i)∩ L(T_↓ (v_1,…,v_i-1),A)∖ A|=b for all i∈[t].

  * For u∉ A, we say that the sequence 
(𝐯|u)=(v_1,…,v_t,u) 
is (A,b)-problematic for T if 
𝐯 is (A,b)-admissible for T and
|N(u)∩ L(T_↓𝐯,A)∖ A|≥ b+1.

  * We say that T is (A,b)-good if there
is no (A,b)-problematic sequence for T.



Note that b-goodness now is the same as (∅,b)-goodness. Hence, Theorem <ref>
follows from the following slightly
more general theorem by setting A=∅.


Let T be a tree with v(T)≥ 2 and A⊂ V(T), then the following equivalence holds:

    .
    [ When Staller starts, Dominator has a strategy to dominate all;                 vertices of V(T)∖ N[A] in the (b:1) game on T;                         by claiming vertices in V(T)∖ A only. ]} ⇔ T is (A,b)-good.




 

Proof of "⇒":
We do a proof by contraposition.
Assume that T is not (A,b)-good. 
Then, by definition, there is an (A,b)-problematic sequence
(𝐯|u) = (v_1,v_2,…,v_t,u) for T.
Note that by definition v_1,v_2,…,v_t,u∉ A.
Staller can play as follows:
In the first round, 
Staller claims v_1, and Dominator is forced
to claim all the b vertices of 
N(v_1)∩ L(T)∖ A,
as otherwise Staller could block one of these vertices
immediately within his next move.
(For this note that all these b vertices
belong to V(T)∖ N[A], since their only neighbour is v_1∉ A.)
Afterwards note that no vertex of 
V(T_↓ (v_1),A)∖ A is dominated yet. Then Staller claims v_2, and analogously Dominator is forced to claim the b vertices of 
N(v_2)∩ L(T_↓ (v_1),A)∖ A.
Staller continues like this and claims v_3,…,v_t,u
in the next rounds. After v_i is claimed for i∈ [t],
Dominator is always forced to claim the b vertices of
N(v_i)∩ L(T_↓ (v_1,…,v_i-1),A)∖ A.
But then, immediately after u is claimed by Staller,
Dominator cannot claim all the at least b+1 vertices
of 
N(u)∩ L(T_↓ (v_1,…,v_i),A)∖ A. 
Then Staller can
claim one of these vertices, say z, in the next round, and Dominator can never dominate this particular vertex z.
By choice of z, it is clear that z∉ A.
Moreover, since N(z)⊂ N({v_1,v_2,…,v_t,u}),
we also have z∉ N(A). Hence, Dominator does not manage
to dominate all of V(T)∖ N[A].




Proof of "⇐":
We prove this direction by strong induction on v(T).
The base case can be checked easily. So, for the induction step (≤ k-1 → k),
consider any tree T with v(T)=k
and any set A⊂ V(T) such that T is (A,b)-good.
The goal is to show that
Dominator has a strategy to dominate all vertices of
V(T)∖ N[A] in the (b:1) game on T
by only claiming vertices in V(T)∖ A and
when Staller is the first player.

Assume that in the first round, Staller claims
some vertex w.
Let T_1,T_2,…,T_ℓ be the components
of T-w.
Moreover, for each i∈ [ℓ], 
let x_i be the unique neighbour of w in T_i,
and let A_i = V(T_i)∩ A. 
We claim that the following statements hold.


Let i∈ [ℓ].
Assume T_i is not 
(A_i,b)-good.
Then there exists an
(A,b)-admissible sequence
𝐰 for T,
contained in V(T_i),
such that
x_i∈ N(w) ∩ L(T_↓𝐰,A)∖ A.




For every i∈ [t], 
T_i is (A_i∪{x_i},b)-good.


Before we prove the above claims, let us explain first
how the induction can be completed.

By Claim <ref> it follows that
at most b components T_i are not
(A_i,b)-good. Indeed, for contradiction assume that this was wrong, and w.l.o.g. say that T_i 
is not (A_i,b)-good for every i∈ [b+1].
Then let 𝐰_i be
an (A,b)-admissible sequence for T
as promised by Claim <ref>,
for each i∈ [b+1].
The concatenation 
𝐰 = 𝐰_1 ∘𝐰_2 ∘…∘𝐰_b+1 then is an (A,b)-admissible sequence for T
such that each of the vertices x_i with i∈ [b+1]
belongs to N(w)∩ L(T_↓𝐰,A)∖ A. Hence, (𝐰|w) is an
(A,b)-problematic sequence for T,
in contradiction to T being (A,b)-good.

In particular, by possibly reordering the components T_i, we can assume that there is some r∈{0,1,…,b}
such that T_i is (A_i,b)-good if and only if i > r.
Dominator's strategy for the first round then is to claim x_1 (even if T_1 is already (A_1,b)-good) and then claim the vertices
x_2,x_3,…,x_r (and skip his remaining elements if r<b). Note that this is allowed since
by Claim <ref> we have
x_1,x_2,…,x_r∉ A.
 
For the next rounds, Dominator always plays on the same component T_i
on which Staller made his previous move.
By induction hypothesis and using Claim <ref>, for every i∈ [r],
Dominator can dominate in T_i every vertex of V(T_i)∖ N[A_i∪{x_i}]
without claiming elements of A_i∪{x_i}.
Together with the element x_i from the first round,
Dominator thus manages to dominate
V(T_i)∖ N[A].
Similarly, for every i>r, since T_i is (A_i,b)-good,
Dominator can dominate in T_i every vertex of V(T_i)∖ N[A] without claiming vertices in A_i. As w is dominated by x_1, Dominator hence dominates V(T)∖ N[A] completely.




Hence, it remains to prove Claim <ref>
and Claim <ref>.



Proof of Claim <ref>:
By assumption
there exists an (A_i,b)-problematic sequence 
(𝐯|u) for T_i.



Case 1: First, We consider the case that
there exists an (A_i,b)-problematic sequence 
(𝐯|u) for T_i
such that
x_i∈ V(T_↓𝐯,A_i).
Then 𝐯 is an (A,b)-admissible sequence 
for T. However, since T is (A,b)-good,
we know that (𝐯|u) is not 
(A,b)-problematic in T.
Therefore, u has
at most b neighbours in L(T_↓𝐯,A)∖ A. But then, since
u has at least b+1 neighbours in 
L((T_i)_↓𝐯,A_i)∖ A_i,
and since
L((T_i)_↓𝐯,A_i)∖ A_i
⊂ 
(L(T_↓𝐯,A)∖ A) ∪{x_i}
holds, 
it immediately follows that
N(u) ∩ L((T_i)_↓𝐯,A_i)∖ A_i
has size b+1 and contains x_i,
while
|N(u)∩ L(T_↓𝐯,A)∖ A|=b.
Now, setting 𝐰=(𝐯|u),
we see that 𝐰 is an
(A,b)-admissible sequence for T,
contained in V(T_i),
such that
x_i∈ N(w) ∩ L(T_↓𝐰,A)∖ A.



Case 2: We next consider the case that 
x_i∉ V(T_↓𝐯,A_i)
for every
(A_i,b)-problematic sequence 
(𝐯|u)=(v_1,…,v_t,u) for T_i.

We first observe that for every such
sequence, we have x_i ∉{v_1,…,v_t,u},
since otherwise 𝐯 would be an
(A,b)-admissible sequence for T
with |N(u)∩ L(T_↓𝐯,A)∖ A|≥ b+1, contradicting that T is (A,b)-good.
Hence, since x_i∉ V(T_↓𝐯,A_i), there must be some j∈ [t] such that 
x_i∈ N(v_j)∩ 
L(T_↓ (v_1,…,v_j-1)A_i)∖ A_i.
Among all possible
(A_i,b)-problematic sequences for T_i,
we now choose (𝐯|u)
such that k:=t-j is minimal.

Having (𝐯|u) fixed, 
we define an auxiliary digraph D_𝐯 
on {v_j,…,v_t,u} as follows: 
For simplicity write v_t+1:=u,
and for every j≤ j_1<j_2 ≤ t+1 
we put a directed edge (v_j_1,v_j_2) in D_𝐯
if and only if

    |N(v_j_2)∩ L(T_↓ (v_1,…,v_j_1),A_i)∖ A_i|
    >
    |N(v_j_2)∩ L(T_↓ (v_1,…,v_j_1-1),A_i)∖ A_i|.

This means, roughly speaking, that in the step when v_j_1 and its b leaf-neighbours are deleted, the vertex v_j_2
receives a new leaf-neighbour.
Note that this can only happen, if there is a (unique) path
(v_j_1,v,v_j_2) in 
L(T_↓ (v_1,…,v_j_1-1),A_i)
and the deletion of v_j_1 makes v a leaf.

The underlying (undirected) graph of D_𝐯
is a forest, and by construction, in D_𝐯
every directed edge goes from a vertex of smaller index to a vertex of higher index. We moreover observe the following three simple properties:


  (a) The vertex u has an ingoing edge.


Indeed, if this was not the case, then
|N(u)∩ L(T_↓ (v_1,…,v_j-1),A_i)∖ A_i|≥ b+1. Let 𝐯' be obtained from 𝐯 by deleting v_j.
Then (𝐯'|u)
would be an (A_i,b)-problematic sequence 
for T_i such that
x_i∈ V(T_↓𝐯',A_i),
in contradiction to the assumption of Case 2.



  (b)
No other vertex than u has out-degree 0.

Indeed, assume instead that we could find
a vertex v_ℓ≠ u with out-degree 0.
Let 𝐯' be obtained from 𝐯 by deleting v_ℓ.
Then (𝐯'|u)
would be an (A_i,b)-problematic sequence 
for T_i, contradicting the minimality of the parameter k.



  (c)
No other vertex than v_j has in-degree 0.

Indeed, assume instead that we could find
a vertex v_ℓ≠ v_j with in-degree 0.
Then the sequence
(v_1,…,v_j-1,v_ℓ,v_j,…,v_ℓ -1,v_ℓ +1,v_t|u)
would be an (A_i,b)-problematic sequence 
for T_i, contradicting the minimality of the parameter k.


From (a)–(c) it follows that
D_𝐯 is the directed path
(v_j,…,v_t+1). Since therefore
every vertex in this path except from v_j 
has exactly one ingoing edge
and since (𝐯|u) is
(A_i,b)-problematic for T_i, 
we conclude  that

    |N(v_ℓ)∩ L(T_↓ (v_1,…,v_j-1),A_i)∖ A_i|=b-1   for every j<ℓ≤ t  ,
        |N(u)∩ L(T_↓ (v_1,…,v_j-1),A_i)∖ A_i|=b  ,

and that in T_↓ (v_1,…,v_j-1),A_i
for every j<ℓ≤ t+1 there must be a (unique) vertex w_ℓ of degree 2
such that (v_ℓ-1,w_ℓ,v_ℓ) forms a path.
But then setting
𝐰:=(v_1,…,v_j-1,u,v_t,…,v_j+1), we have (𝐰 | v_j) is an (A,b)-problematic sequence for T, contained in V(T_i), such that x_i∈ N(w) ∩ L(T_↓𝐰,A)∖ A..






Proof of Claim <ref>:
Assume the statement is wrong.
Then we could find
an (A_i∪{x_i},b)-problematic sequence 
(𝐯|u) for T_i.
This sequence would then also be
(A,b)-problematic for T,
in contradiction to T being (A,b)-good.




This completes the proof of Theorem <ref>.



While Theorem <ref> is a direct consequence of the just proven theorem, we can also conclude the following variant when Dominator is the first player.


Let T be a tree. When Dominator starts, then 
Dominator wins the b-biased Maker-Breaker domination game on T
if and only if there exists A⊂ V(T) of size at most b such that T is (A,b)-good.




If there exists A ⊂ V(T) of size b
such that T is (A,b)-good,
then in his first move, Dominator can occupy A
and afterwards win according to 
Theorem <ref>.

Otherwise, if no such A exists, then no matter which 
set A Dominator claims in the first round, 
T is not (A,b)-good. Hence,
by Theorem <ref> Staller has a strategy
to make sure that Dominator will not dominate everything
in V(T)∖ N[A].







 §.§ Maker-Breaker domination number for trees


For the game in which Dominator starts, note that trivially

    1≤γ_MB(T,b)≤⌈n/b+1⌉

holds for every tree T on n vertices,
provided that γ_MB(T,b)<∞.
At the end of this subsection, Remark <ref> will explain why every integer in this range can be realized by some tree T.
By proving Theorem <ref>, we will however prove a significantly larger lower bound for γ_MB'(T,b), which is linear in n. For the proof, we will make use of the following statement which can be seen as a generalization 
of Theorem 4.4.(i) in <cit.>.


Let T be a tree which is b-good for some b∈ℕ, and let 
𝐯=(v_1,v_2,…,v_t) be 
an (∅,b)-admissible sequence for T.
Then

    γ_MB'(T,b) = v(T)-v(T_↓𝐯)/b+1 + γ_MB'(T_↓𝐯,b).





Let 𝐯=(v_1,v_2,…,v_t) be 
an (∅,b)-admissible sequence for T,
and note that t(b+1)=v(T)-v(T_↓𝐯).

"≥":
In the first t rounds, Staller can claim the
vertices v_1,v_2,…,v_t, always forcing
Dominator to claim b leaf-neighbours (as in the proof of
Theorem <ref>).
Afterwards, the game reduces to T_↓𝐯 where no vertex is claimed yet and no vertex is dominated by Dominator so far.
Hence, Staller can continue the game so that it lasts
at least γ_MB'(T_↓𝐯,b) further rounds,
leading to a total of at least
t+γ_MB'(T_↓𝐯,b)
rounds.



"≤":
For each i∈ [t],
let V_i = {v_i}∪ (N(v_i)∩ L(T_↓ (v_1,…,v_i-1),∅)), and note
that each V_i induces a star with b edges.
Now, Dominator can play according to the following strategy: Whenever Staller claims a vertex in some V_i,
Dominator claims all remaining vertices in V_i, 
and this way dominates all of V_i.
Whenever Staller plays on V(T_↓𝐯)=V∖ (⋃_i∈ [t] V_i) and Dominator does not dominate V(T_↓𝐯) yet,
Dominator plays on T_↓𝐯 with the strategy that ensures to win within γ_MB'(T_↓𝐯) rounds. 
Whenever Staller plays on V(T_↓𝐯) while 
Dominator already dominates V(T_↓𝐯),
then Dominator claims
b vertices of a set V_i which is not yet dominated.
Following the strategy, Dominator wins within
t+γ_MB'(T_↓𝐯,b) rounds.


With the above lemma in hand, we can now prove Theorem <ref>.




(a) The upper bound is trivial as the game cannot last longer than ⌈n/b+1⌉ rounds.
Hence, we focus on the lower bound from now on.
If γ(T)≥n/b+3, then the lower bound in (a) holds trivially, since Dominator has less than
n/b+3 vertices as long as 
less than n/b(b+3) rounds are played.
Thus, from now on we can assume that
γ(T) < n/b+3.
In particular, there is a set A⊂ V(T) 
of size |A| < n/b+3
such that A∪ N_T(A)=V(T).
To each w∈ N_T(A) assign a unique vertex
v∈ A such that vw∈ E(T);
and for each vertex v∈ A, denote with
d^*(v) the number of vertices w∈ N_T(A)
which are assigned to v in this way.
Note that

    ∑_v∈ A d^*(v) = |N_T(A)| = n - |A| > (b+2)n/b+3.

Staller's strategy is as follows:
As long as less than |A|/b+1 rounds were played so far, 
Staller claims a free vertex in A for which
d^*(v) is maximal.
Let A'⊂ A be the set claimed this way. Then
|A'|=⌈|A|/b+1⌉ and

    ∑_v∈ A' d^*(v) ≥1/b+1∑_v∈ A d^*(v)
    > (b+2)n/(b+1)(b+3).

We claim that 
T-A' has at least n/b+3 components. To see this,
let c be the number of these components, and
consider the tree T' obtained from T by contracting the components of T-A'. Then v(T')=|A'|+c and hence

    e(T')=v(T')-1=|A'|+c-1 .

Moreover,

    e(T') ≥ e_T(A',V(T)∖ A')
    ≥∑_v∈ A' d^*(v) ≥(b+2)n/(b+1)(b+3)

and therefore

    c ≥(b+2)n/(b+1)(b+3) - |A'| + 1 
        > (b+2)n/(b+1)(b+3) - |A|/b+1
        > (b+2)n/(b+1)(b+3) - n/(b+1)(b+3)
    = n/b+3.

Now note that Dominator needs at least one vertex in each component of T-A' to win the b-biased Maker-Breaker domination game on T. As she can touch at most b of these components in each round, she needs to play at least n/b(b+3) rounds.



(b) 
Let f(b):=(⌊b/2⌋ + 1)(⌈b/2⌉ + 1). The main part of the proof will be the following claim.


For all integers
k≥ 1 and b≥ 2, there exists a tree T_k,b such that
γ_MB'(T_k,b) = ⌈k/f(b)⌉.


	

Before we prove this claim, let us first explain how the statement (b) can be concluded.
Having n and b fixed, 
for any 0≤ s≤⌊n/b+1⌋
consider any tree T(s) on n vertices for which there is an (∅,b)-admissible sequence 𝐯=(v_1,…,v_s)
with T_↓𝐯 = T_n-s(b+1),b;
an example is illustrated in Figure <ref>.
Then, using Lemma <ref> 
and the claim above we obtain

    γ_MB'(T(S),b)=
    s + γ_MB'(T_n-s(b+1),b,b)=
    s + ⌈n-s(b+1)/f(b)⌉ .

Now for the expression F(s):=s + ⌈n-s(b+1)/f(b)⌉ observe that
F(0)=⌈n/f(b)⌉,
and
F( ⌊n/b+1⌋ ) = ⌈n/b+1⌉,
and
F(j)≤ F(j+1)≤ F(j)+1
for every 0≤ j≤⌊n/b+1⌋, since
b+1/f(b)≤ 1.
Hence, taking for s all integer values
in the range [0,⌊n/b+1⌋],
we see that F(s) takes all integer values in the range [ ⌈n/f(b)⌉, ⌈n/b+1⌉ ], proving part (b) of the theorem.



It remains to prove Claim <ref>, which will be done in the following.

For short, let x:=⌈b/2⌉
and q:= ⌈k/x+1⌉.
We set r:=k-(q-1)(x+1) and note that
1≤ r≤ x+1.
Moreover, for later arguments note that

    ( ⌈k/f(b)⌉-1)
    ·( b-x+1)
    <
     k/f(b)·( ⌊b/2⌋ + 1)
    =
    k/x+1.


In order to construct the tree T_k,b,
we proceed as follows: we first take q disjoint stars
S_1,S_2,…,S_q with centers z_1,z_2,…,z_q
such that e(S_i)=x for every i∈ [q-1]
and e(S_q)=r-1, and then we we add the edges z_iz_i+1 with i∈ [q-1];
an illustration is given in Figure <ref>.
Note that the resulting graph has ∑_i∈ [q] v(S_i)=k
vertices.

	

We claim that γ_MB'(T_k,b) = ⌈k/f(b)⌉. For this we distinguish the cases r≠ 1 and r=1.



Case 1: Assume that r≠ 1.

We first prove that 
γ_MB'(T_k,b) ≤⌈k/f(b)⌉.
To do so, let Dominator play as follows:
If Staller in his previous move claimed a vertex of a star S_i, i∈ [q], which does not contain any vertices claimed by Dominator, then Dominator first claims all remaining vertices
of S_i (in total at most x vertices),
and then she claims as many centers z_i as possible
such that the corresponding stars S_i do not contain a vertex of Staller, preferring i<q. 
This way, in each round Dominator makes sure to dominate all vertices of the just mentioned stars, and the number of these vertices is at least
(b-x+1)(x+1) = f(b), except for maybe the last round. It follows that Dominator
claims a dominating set of T_k,b within at most 
⌈k/f(b)⌉ rounds.

Next, we prove
γ_MB'(T_k,b) ≥⌈k/f(b)⌉.
To do so, consider the following strategy for Staller.
In the first ⌈k/f(b)⌉-1 rounds,
Staller distinguishes three cases:
(1) If Staller can claim a vertex in such a way that he can complete the closed neighbourhood of a vertex, then he does so and wins the game. (2) If such a vertex does not exist, Staller claims a free star center z_i with i<q. (3) Otherwise, if such a center also does not exist, Staller forfeits the game (we will see in a moment that this case cannot happen).
In round ⌈k/f(b)⌉ Staller 
again plays according to (1) if possible, and otherwise
he plays arbitrarily.



In order to show that Staller succeeds
in making the game last at least ⌈k/f(b)⌉
rounds, assume first that for some j≤⌈k/f(b)⌉-2 Staller could follow his strategy and so far there was no option for following case (1). We now explain why then Staller can play his (j+1)-st move according to (2):
Let z_i_1,z_i_2,…,z_i_j be the star centers that Staller claimed in the first j rounds.
Since (1) did not happen yet, Dominator must have claimed
all the remaining vertices of the corresponding stars
S_i_1,S_i_2,…,S_i_j,
in total jx vertices. As so far Dominator has claimed
jb vertices in total, 
next to S_i_1,S_i_2,…,S_i_j
she can have touched at most j(b-x) further stars,
leading to a total of at most 

    j(b-x+1) 
    ≤( ⌈k/f(b)⌉-2)
    ·( b-x+1)
    (<ref>)<k/x+1 
    - (b-x+1)
    ≤ q - 1

stars that may contain vertices of Dominator.
In particular, there must be a free center z_i with i<q
so that Staller can follow (2). Thus, Staller can always follow
the proposed strategy without forfeiting the game.

Now, after at the end of round ⌈k/f(b)⌉ - 1, the same argument as above gives that Dominator can have touched at most

    ( ⌈k/f(b)⌉-1)
    ·( b-x+1)
    (<ref>)<k/x+1≤ q

stars so far, and hence there must be at least one star
where none of its leaves is yet dominated.
So, Dominator needs to play at least the round 
⌈k/f(b)⌉
before she can win.




Case 2:  Assume that r = 1. The proof is essentially the same as in Case 1, but since S_q is a single vertex,
Staller can force Dominator to claim z_q by claiming the center z_q-1. In other words, we can instead from now on redefine our construction so that z_q belongs to S_q-1, and hence
e(S_q-1)=x+1 and S_q does not exist.

For proving that 
γ_MB'(T_k,b) ≤⌈k/f(b)⌉,
let Dominator play as in Case 1.
In all rounds (except maybe the last round)
in which Staller does not claim z_q-1,
Dominator can again dominate b-x+1 stars completely,
and hence dominates (b-x+1)(x+1)=f(b) vertices of these stars.
In a round (except maybe the last round)
in which Staller claims z_q-1,
Dominator must claim the x+1 leaves of S_q-1
and can then care about further b-x-1 stars by claiming their centers; the dominated vertices in these considered stars are then
(b-x-1)(x+1)+(x+2) = f(b)-x. 
Assume that Dominator would not win the game within the first 
⌈k/f(b)⌉ rounds, then with the same argument as above
she would still have dominated at least

    ( ⌈k/f(b)⌉ - 1 )f(b) + 1· (f(b)-x)
    =
    ⌈k/f(b)⌉· f(b) - x

vertices by then. Now, note that by the assumption of the case we have
k≡ 1 (mod x+1) and thus, since f(b) is a multiple of x+1, we get k≡ t(x+1)+1 (mod f(b)) for some integer t.
In particular, the number of dominated vertices can be further estimated from below by

    ⌈k/f(b)⌉· f(b) - x
    ≥k+x/f(b)· f(b) - x = k,

in contradiction to the assumption that Dominator has not won already.



For proving that 
γ_MB'(T_k,b) ≥⌈k/f(b)⌉,
let Staller also play as in Case 1,
except that in the first round he claims the center z_q-1
of the biggest star S_q-1.
Due to the additional leaf that Dominator needs to claim
in the first round when compared with Case 1,
Dominator looses one move (one vertex) for claiming a center of another star. Because of this, all the estimates in (<ref>) and (<ref>)
can be changed by adding the summand -1.
In particular, in (<ref>) we then get that
Dominator has touched less than q-1 stars within the first
⌈k/f(b)⌉ - 1 rounds, and hence at least one more round is needed for her to win the game.





For the case when Dominator starts the game, γ_MB(T,b) can take every integer value t between 1 and n/b+1. For this consider the tree T obtained as follows. Take t disjoint stars S_1,…,S_t such that e(S_i)=b for every i<t and e(S_t)=n-(b+1)t, and add a path through the centers of these stars; then γ_MB(T,b)=t. We leave the details to the reader.






 §.§ Number of dominated vertices

Finally, we prove Theorem <ref>.


For the first part consider ℱ = { N[v]:  v∈ V(G)}
and note that 

    ∑_F∈ℱ (1+b)^-|F|≤v(G)/(b+1)^2 .

By Theorem <ref>,
Dominator (playing as Breaker) can make sure that Staller claims at most
v(G)/(b+1)^2 sets of ℱ completely.
This in turn implies the first part of the theorem.

For the sharpness, let n∈ℕ and consider
a graph on n vertices obtained as follows:
take b+1 stars each of size ⌊n/b+1⌉,
and add a path through the centers of these stars.
After the first move of Dominator, there is at least one star
which was not touched. Staller first claims the center,
and afterwards roughly a 1/b+1-fraction of its leaves.
These leaves, the number of which is roughly n/(b+1)^2,
are not dominated by Dominator.


















§ MINIMUM DEGREE CONDITION


In this section, we will make use of the following simple observation which can also be found in <cit.>.


Let b∈ℕ, 
and let T be a perfect (b+1)-ary tree,
and let  = (T) be the hypergraph 
on vertex set V(T) where
the vertices of any root-leaf path of T 
form a hyperedge of . Then, Maker
wins the (1:b) Maker-Breaker game on (X,),
provided she is the first player.


Having this result and Beck's Criterion in our hand, we can prove our first minimum degree condition (Theorem <ref>) for Maker-Breaker domination games.


For the first part of the theorem, let G be any graph on n vertices with δ(G) > log_b+1(n) - 1. 
Consider ℱ = { N[v]:  v∈ V(G)}
and note that 

    ∑_F∈ℱ (1+b)^-|F|≤ n (1+b)^-(δ(G)+1) < n (1+b)^-log_b+1(n) = 1  .

Hence, by Theorem <ref>
it follows that Dominator (playing as Breaker) can claim an element in each set of ℱ. That is, Dominator claims a vertex in each closed neighbourhood, so that all of her vertices form a dominating set.

For the second part, let T_k be a perfect (b+1)-ary tree 
with k+1 levels (the root is the first level), 
and let G be the graph obtained from taking (b+1) disjoints
copies of T_k and adding additional edges between every vertex
and all its descendants.
Then n:=v(G)= (b+1)∑_i=0^k (b+1)^i < (b+1)^k+2 and
δ(G) = k > log_b+1(n)-2.
We claim that Dominator loses the (b:1) Maker-Breaker domination game on G.
For this, Staller plays as follows:
After Dominator's first move, where she claims b vertices,
there will be at least one copy of T_k untouched.
On this copy, Staller follows Maker's strategy guaranteed by Lemma <ref>, so that Staller claims all the vertices of a root-leaf path. The leaf of this root-leaf path cannot be dominated by Dominator.



Next we prove Theorem <ref>
by combining a simple probabilistic argument 
and Beck's Criterion.


At first observe that there exists a
subset A⊂ V(G)
such that

    |A|≤10nln(n)/δ(G)  and   
    |N(v)∩ A| > 2ln(n) 
      for every  v∈ V(G).

The existence can be proven with a standard probabilistic argument.
Indeed, let A⊂ V(G) be a random subset where
every vertex of V(G) is contained in A independently
with probability p=9ln(n)/δ(G).
Then 𝔼(|A|)=np=9nln(n)/δ(G) and, using a Chernoff inequality (Lemma <ref>), we have

    Prob[ |A|> 10nln(n)/δ(G)]
    ≤exp( - 3nln(n)/δ(G))
    ≤exp( - 3ln(n) ) = n^-3.

Moreover, for every vertex v we have
𝔼(|N(v)∩ A|)=|N(v)|p ≥ 9ln(n) and
thus, using a Chernoff inequality (Lemma <ref>), we conclude

    Prob[ |N(v)∩ A| ≤ 2 ln(n) ]
    ≤exp( - 2ln(n) ) = n^-2.

Thus, by taking a union bound, we see that
(<ref>) fails with probability at most
n · n^-2 + n^-3 <1.



Now, fix such a set A and consider
ℱ := {N(v)∩ A:  v∈ V(G)}.
Then

    ∑_F∈ℱ (1+b)^-|F|≤ n (1+b)^-2ln(n) < 1

for every b≥ 1 and n≥ 2.
Thus, using Beck's Criterion (Theorem <ref>), we see that by playing on A
only, Dominator (playing as Breaker) can get a vertex in each of the sets A∩ N(v). As she has an element in each N(v) then, she has claimed a dominating set.
Moreover, for playing on A with bias b, Dominator only needs at most |A|/b rounds, giving the desired bound.




















§ CONCLUDING REMARKS AND OPEN PROBLEMS



Dominating trees.
In part (a) of Theorem <ref> we provide
general bounds on γ_MB'(T,b)
for any tree T, and with part (b) we  show that our general lower bound is optimal up to a multiplicative factor of at most 4. Naturally this leads to the question of finding a tight lower bound and a characterization of extremal trees.


Given integers b and n.
What is smallest possible value
γ_MB'(T,b)
among all trees T on n vertices?



Given integers b and n.
Characterize all trees T on n vertices,
which minimize γ_MB'(T,b).


While part (b) of Theorem <ref> also ensures the existence of a tree T
such that γ'_MB(T,b)=t whenever 
⌈n/f(b)⌉≤ t ≤⌈n/b+1⌉ with
f(b):=(⌊b/2⌋ + 1)(⌈b/2⌉ + 1) holds, we do not a have a general approach
to determine γ'_MB(T,b) or 
γ_MB(T,b) for any given tree T.
In particular, we ask the following.


Is there a polynomial time algorithm that finds
γ'_MB(T,b) or γ_MB(T,b) for any given tree T and positive integer b?




Dominating random graphs.
In order to obtain a better understanding of biased
Maker-Breaker domination games, it also seems natural to
study such games when played on randomly generated graphs.

For instance, let G∼𝒢_2(n,r) be a random geometric graph, which is obtained by placing n points uniformly at random into the square [0,1]^2 and by adding an edge between any two points if their Euclidean distance is at most r. Then, with a simple pairing strategy
together with an argument similar to the proof of
Theorem 2 in <cit.>,
it can be shown that

    γ_MB(G,b), γ'_MB(G,b) = ( 2√(3)/9b + o(1) ) r^-2

holds a.a.s.
Moreover, if we instead consider the Erdős-Rényi random graph model G_n,p with constant p,
a simple randomized strategy for Dominator can be
used in order to show that a.a.s.

    γ_MB(G,b), γ'_MB(G,b) = (1+o(1)) 1/blog_1/(1-p)(n)

holds. 
That is, in both of the above cases the number of elements that Dominator a.a.s. needs to claim until winning the game on G is asymptotically equal to the domination number γ(G). We wonder whether even equality holds.


 Let an integer b≥ 1 be given, and let
 G∼𝒢_2(n,r) for some r∈ (0,1).
 Is it true that 
 γ_MB(G,b) = γ'_MB(G,b) = ⌈γ(G)/b⌉
 holds a.a.s.?



 Let an integer b≥ 1 be given, and
 let G∼ G_n,p for some p∈ (0,1).
 Is it true that 
 γ_MB(G,b) = γ'_MB(G,b) = ⌈γ(G)/b⌉
 holds a.a.s.?


Note that if the answer to the last question is yes,
then γ_MB(G,b) and γ'_MB(G,b)
would be concentrated on two values if G∼ G_n,p, see <cit.>.
But even if the answer is no, it would be interesting to
analyse whether concentration results
similar to those in <cit.>
can be proven.



Variants. For all questions discussed in this paper, it also seems natural to consider variants of Maker-Breaker domination games, for example where Dominator's goal is to claim a total domination set or a k-dominating set for any positive integer k.







§ ACKNOWLEDGEMENT

Parts of this paper were obtained in the Bachelor thesis of the first author at the Hamburg University of Technology. The first author wants to thank Anusch Taraz and his research group
for the supervision.






amsplain








 Paper Title: On the asymptotic enumeration and limit shapes of monotone grid classes of permutations.






































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smallmx[1][] ([ )









On the asymptotic enumeration and limit shapes 
of monotone grid classes of permutations
    ^†Noura Alshammari^† and David Bevan^†
    
========================================================================================




^†Department of Mathematics and Statistics, The University of Strathclyde, Glasgow, Scotland.



2020 Mathematics Subject Classification:
05A05, 

05A16. 




§ ABSTRACT 
We exhibit a procedure to asymptotically enumerate monotone grid classes of permutations.
This is then applied to compute the asymptotic number of permutations in any connected one-corner class.
Our strategy consists of
enumerating the gridded permutations,
finding the asymptotic distribution of points between the cells in a typical large gridded permutation,
and analysing in detail the ways in which a typical permutation can be gridded.
We also determine the limit shape of any connected monotone grid class.





§ INTRODUCTION


We consider a permutation of length n (an n-permutation) to be a linear ordering of the numbers 1,2,…,n, and usually identify a permutation σ=σ_1…σ_n with its plot, the set of points (i,σ_i) in the Euclidean plane.
The monotone grid class (M) is a set of permutations defined by a gridding matrix M whose entries are drawn
from {,,}.
This matrix specifies the permitted shape for plots of permutations in the class.
Each entry of M corresponds to a cell in an M-gridding of a permutation.
If the entry is , any points in the cell must form an increasing sequence;
if the entry is , any points in the cell must form a decreasing sequence;
if the entry is , then the cell must be empty.
A permutation can have more than one M-gridding.
See Figure <ref> for an example.
Formal definitions are given below.

The study of individual grid classes goes back to the work of Stankova <cit.> and Atkinson <cit.> in the 1990s on the skew-merged permutations.
However, monotone grid classes were formally introduced (under a different name) in 2003 by Murphy and Vatter <cit.>.
Huczynska and Vatter <cit.> were the first to call them “grid classes” and to use the notation (M).

Exact enumerative results for monotone grid classes are rare.
A few individual classes have been enumerated, such as the skew-merged permutations (2-1,11,-1)
whose generating function was determined by Atkinson <cit.>.
However, only two general results are known.
The first of these concerns skinny grid classes, which are those whose matrices have a single row.
That is, (M) is skinny if M is simply a vector over {,}.
In the second author's PhD thesis <cit.>,
an iterative procedure is presented which yields the generating function of any skinny monotone grid class, a result which was later generalized by Brignall and Sliačan in <cit.> to skinny classes in which the contents of a single cell may be non-monotone.
The other general result concerns polynomial grid classes (classes with growth rate equal to 1).
The monotone grid class (M) is polynomial if M has at most one  or  in any column or row.
Homberger and Vatter <cit.> describe an algorithm which can enumerate any such class.
Extending either of these general exact enumeration results seems very challenging.
Perhaps the Combinatorial Exploration algorithmic framework described in <cit.> can be adapted to do so.



Given the difficulty of exact enumeration, our goal is the asymptotic
enumeration of monotone grid classes.
In <cit.>, the second author proves that the exponential growth rate,

    ((M)) |_n(M)|^1/n,

 of (M) exists and is equal to the square of the spectral radius of a certain graph 
associated with M.
A simpler proof of (a more general version of) this result is given by Albert and Vatter <cit.>.
Thus we know that

    |_n(M)|  ∼ θ(n) g^n ,

where g is the growth rate of the class and θ(n) is subexponential; that is, θ(n)^1/n=1.
We write f(n)∼ g(n) to denote that f(n)/g(n)=1.


Our primary contribution is a procedure to determine the asymptotic enumeration of a monotone grid class.
This is first applied to skinny classes (Theorem <ref>).
We then establish the asymptotic distribution of points between cells in a typical large gridded permutation (Theorem <ref>),
and use this to
determine grid class limit shapes (Theorem <ref>).
Building on these results, the major part of our work then concerns the asymptotic enumeration of connected one-corner classes (Theorem <ref>).
These are either -shaped, -shaped or -shaped.
See Section <ref> for the relevant definitions.

We begin with a formal definition of an M-gridding.
In this definition, in order to match the way we view permutations graphically, we index matrices from the lower left corner, with the order of the indices reversed from the normal convention.
Thus, 
M_2,1 is the entry that is in the second column from the left and in the bottom row of M.

If M is a gridding matrix with t columns and u rows, then formally an M-gridding of a permutation σ of length n
is a pair of integer sequences
0=c_0 c_1… c_t=n (the column dividers)
and 0=r_0 r_1… r_u=n (the row dividers)
such that
for every column i and row j,
the subsequence of σ with indices in (c_i-1,c_i] and values in (r_j-1,r_j] is
increasing if M_i,j=,
decreasing if M_i,j=, and
empty if M_i,j=.
For example, in the leftmost gridding in Figure <ref>, we have c_1=2, c_2=6 and r_1=1.
If the gridding matrix is clear from the context, we may just talk of a gridding of a permutation.

In practice, it is more natural to consider the assignment of points to cells that is induced by an M-gridding.
With this perspective, we regard the row and column dividers to be the horizontal and vertical lines at half-integer positions that partition the plot of a permutation into cells.
We follow this convention.
With a minor abuse of terminology, we often refer to the matrix entries themselves as cells, calling  entries blank cells, and  and  entries non-blank cells.

The grid class (M) then consists of those permutations that have at least one M-gridding.
We use _n(M) to denote the set of permutations of length n in (M).
A permutation together with one of its M-griddings is called an M-gridded permutation (or just a gridded permutation if the gridding matrix is clear from the context).
The gridded class, consisting of all M-gridded permutations, is denoted by (M),
and the set of M-gridded permutations of length n by _n(M).
When we use σ^# to denote a gridded permutation, its underlying permutation is σ.

Observe that if a gridding matrix M has dimensions t× u, then no permutation in _n(M) can have more than (n+1)^t+u-2 griddings,
because there are only n+1 possible positions for each row and column divider.
Since this is polynomial in n,
the exponential growth rate of the gridded class is equal to that of the grid class <cit.>. 

Building on this observation,
our strategy for the asymptotic enumeration of (M) consists of the following five steps:

  
  * 
  Find the proportion of points that occur in each cell in a typical large M-gridded permutation. 

  
  * 
  Determine the asymptotic enumeration of the corresponding
  gridded class:
  
    |_n(M)|  ∼ θ^#(n) g^n ,

  where g is the exponential growth rate of the class, and θ^#(n) is subexponential.


  * 
  Determine, for each ℓ1, how a typical large M-gridded permutation σ^# must be structured so that its underlying permutation σ has exactly ℓ distinct M-griddings.

  
  * 
  By combining steps <ref> and <ref>, calculate, for each ℓ1, the asymptotic probability
  
    P_ℓσ has exactly ℓ distinct M-griddings : σ^#∈_n(M) .


  
  * 
  Let
  
    κ_M
      ∑_ℓ1P_ℓ/ℓ|_n(M)|/|_n(M)|.

  Then |_n(M)|∼κ_M θ^#(n) g^n.


3
The rest of the paper is structured as follows:

  
  * In Section <ref>, we consider skinny classes.
    We determine how points can “dance” between the cells in a typical large gridded permutation.
     Hence, we deduce the asymptotic enumeration of any skinny class (Theorem <ref>).
  
  * Section <ref> concerns the distribution of points between the cells in a typical large gridded permutation in a connected class (Theorem <ref>).
      This result is then used to determine the shape of a typical permutation in such a class (Theorem <ref>).
  
  * In Section <ref>, we apply our strategy to connected one-corner classes.
      We begin by finding the proportion of points in each cell in a typical large gridded permutation (Section <ref>).
      Then we determine the asymptotic enumeration of gridded connected one-corner classes (Proposition <ref>).
      The heart of this section consists of a detailed analysis of how typical permutations in these classes can be gridded, based on an examination of the eleven possible corner types (Sections <ref>–<ref>).
      This then yields the asymptotic enumeration of any connected one-corner class (Theorem <ref>).
      We conclude by briefly considering the application of our approach to classes with more than one corner (Section <ref>).





§ SKINNY CLASSES


We begin with the simplest case, for which each of the five steps in the analysis is easy.
Recall that (M) is skinny if M is simply a / vector.
Although, as mentioned above, a procedure is known which gives the generating function of any skinny class, this would be a very inefficient way of determining their asymptotic enumeration.




We begin by exactly enumerating skinny gridded classes. 


    If (M) is a k-cell skinny grid class, then |_n(M)|=k^n.


  Any M-gridded permutation can be uniquely constructed from the empty M-gridded permutation by repeatedly adding a new maximum point.
  This point may be placed in any one of the k cells.
  Moreover, there is only one way of adding a maximum to any particular cell, because of the monotonicity constraints.
  See Figure <ref> for an illustration.


6



 §.§ Peaks and peak points


We now introduce some concepts that are important when analysing how permutations can be gridded.
Firstly, given a skinny grid class (M), where M is the /
vector (m_1,…,m_k),
we say that a pair (m_i,m_i+1) of adjacent cells forms a peak if m_i+1≠ m_i.
That is, a peak either looks like 21,-1, which we say points up, or else looks like or 2-1,1, which we say points down.
For example, (51,-1,1,1,-1) has three peaks, two pointing up and one pointing down.

Secondly, suppose we have a skinny grid class (M) with a 
peak Λ,
and that σ^# is an M-gridded permutation with at least two points in each cell, witnessing the orientation of the cell.
Then the peak point of Λ is the highest of the points of σ^# in the two cells of Λ if Λ points up,
and is the lowest of the points of σ^# in 
Λ if Λ points down.
For example, in the rightmost two gridded permutations in Figure <ref>, the peak point (of the only peak in the class) is circled.

Note that our use of the term “peak” is nonstandard, differing from its traditional use to denote a consecutive 132 or 231 pattern in a permutation.
For us a peak is part of a gridding matrix.
On the other hand, a peak point is the central point of either a peak or valley (in the traditional sense) in a gridded permutation.
These concepts are generalised further below, when we consider non-skinny classes.






 §.§ Dancing and constrained gridded permutations



Suppose (M) is skinny and
σ^#∈(M).
If Q is a peak point of σ^#, we say that Q can dance if
Q is immediately adjacent to a

column divider.
The movement of this divider to the other side of Q results in another valid M-gridding of σ.
We say that this new gridded permutation is the result of Q dancing.
Conceptually, we consider Q as dancing from one cell to an adjacent one (although in fact it is the divider rather than the point that moves).
See Figure <ref> for an illustration.
The notion of dancing is generalised below in Section <ref>, when we consider non-skinny classes.

The key to our approach is the fact that
for most permutations the gridding possibilities are heavily restricted.
To 
formalise this observation, we focus our attention on certain well-behaved gridded permutations in which only peak points can dance.
Suppose (M) is skinny and σ^#∈(M). 
We say that σ^# is M-constrained (or just constrained) if

  
  (a) every M-gridding of its underlying permutation σ is the result of zero or more peak points of σ^# dancing,
  and
  
  (b) in every M-gridding of σ, each 
  cell contains at least two points.


For constrained gridded permutations, counting possible griddings is easy.


    If (M) is a skinny grid class with p peaks, and σ^#∈(M) is M-constrained, then σ has exactly 2^p distinct M-griddings.


Since σ^# is constrained, every M-gridding of σ is the result of zero or more peak points of σ^# dancing.
Since each cell of σ^# contains at least two points, σ^# has p distinct peak points (one for each peak), which can dance independently.
For each peak point of σ^#, we can choose whether it dances or not,
yielding a total of 2^p distinct M-griddings for σ.


The following proposition gives sufficient conditions for a gridded permutation in a skinny class to be constrained.


  Suppose (M) is skinny and σ^#∈(M) is such that
  each 
  cell contains at least four points and
  there are no two adjacent cells whose contents together form an increasing or decreasing sequence.
  Then σ^# is M-constrained.


The contents of each cell of σ^# consists of an increasing or decreasing sequence of 
points.
However, there is no pair of adjacent cells whose contents together form an increasing or decreasing sequence.

Thus, in any M-gridding of σ, by the monotonicity constraints, there must be
a divider between each pair of adjacent cells of σ^# that have the same orientation
and also
a divider adjacent to each peak point of σ^#.
So any M-gridding of σ can be formed from σ^# by zero or more of its peak points dancing.

Moreover, in any M-gridding of σ,
at most two points from any cell of σ^# (the first and the last) can be gridded in another cell.
So each cell of any M-gridding of σ contains at least two points.


Constrained gridded permutations are of interest not only because it is easy to count their griddings, but also because almost all gridded permutations are constrained.


If (M) is skinny, then almost all M-gridded permutations are M-constrained:

    σ^# is M-constrained : σ^#∈_n(M) 1 .



  Suppose M has k cells.
  The number of n-point M-gridded permutations with exactly m points in a given cell equals
  
    nm(k-1)^n-m <  n^m(k-1)^n .

  Here, nm is the number of ways of choosing the points in the given cell, and (k-1)^n-m is the number of ways of distributing the remaining points.
  So the total number of n-point M-gridded permutations with fewer than four points in some cell
  is less than 4kn^3(k-1)^n,
  there being four choices for the value of m,
  and k choices of cell.

  Similarly, the number of n-point M-gridded permutations with a given pair of adjacent cells forming an increasing or decreasing sequence of length m is less than
  
    (m+1)(k-1)^n  <  (n+1)(k-1)^n .

  Here,
  (k-1)^n is an upper bound on the number of ways of distributing the points if we consider the pair of adjacent cells merged to form a single “super cell”, 
  and the factor m+1 is the number of choices for the position of the divider that splits the pair of cells.
  So the total number of n-point M-gridded permutations with two adjacent cells forming an increasing or decreasing sequence
  is less than (n+1)^2(k-1)^n+1,
  there being n+1 choices for the value of m,
  and k-1 choices for the pair of cells.

  Thus, by Propositions <ref> and <ref>, the proportion of n-point M-gridded permutations which are not constrained is less than
  
    4kn^3(k-1)^n  +  (n+1)^2(k-1)^n+1/k^n < 
      4kn^3(1-1/k)^n,

  which converges to zero as n tends to infinity.


The asymptotic enumeration of skinny grid classes then follows directly from Propositions <ref>, <ref> and <ref>.


    If (M) is a k-cell skinny grid class with p peaks, then |_n(M)| ∼ 2^-pk^n.


For almost all of the k^n distinct M-gridded permutations, the underlying permutation has exactly 2^p M-griddings.





§ THE DISTRIBUTION OF POINTS BETWEEN CELLS


As we have seen, almost every large permutation in a given skinny class has the same number of griddings (that depends only on the number of peaks).
In non-skinny classes this is not the case:
the number of griddings may depend on the structure of the permutation.
In order to determine the asymptotic probability that a permutation has a specific number of griddings,
we need to know the proportion of points that occur in each cell in a typical large gridded permutation.
This is the focus of this section.

Much of the analysis required can be found in the paper of Albert and Vatter <cit.> and (in a more sketchy form) in the second author's PhD thesis <cit.>.
However, neither of these works contains all that we need, so we reproduce the argument in full here.
Our presentation combines ideas from both approaches.

We begin by introducing a family of matrices to record the number of points or proportion of points in each cell (Section <ref>).
We then determine the asymptotic number of gridded permutations with a given distribution (Section <ref>).
This is followed by calculating the distribution of points for which this enumeration is the greatest (Section <ref>).
We then prove 
that this distribution is unique if the class is connected (Section <ref>).
Finally, we establish in Theorem <ref> that the distribution of points in almost all gridded permutations
is close to this maximal distribution, 
This is then used to determine the limit shapes of connected classes (Theorem <ref>). 




 §.§ M-admissible and M-distribution matrices


Given a gridding matrix M, we say that a nonnegative real matrix A of the same dimensions as M is M-admissible if A_i,j is zero whenever M_i,j is blank.
We refer to the sum of the entries of such a matrix as its weight.

We use integer M-admissible matrices to record the number of points in each cell.
Suppose A=(a_i,j) is an integer M-admissible matrix of weight n.
Then _A(M) denotes the  of _n(M) consisting of those M-gridded permutations with a_i,j points in cell (i,j), for each (i,j).
For example, the leftmost gridded permutation in Figure <ref> on page figGriddings is an element of 

    _2   3   3
    0   1   0(3-1,-1,10,1) .


Thus _n(M) can be partitioned into subsets as follows:

    _n(M) _A=n_A(M),

where the disjoint union is over all integer M-admissible matrices of weight n.
The number of gridded permutations in one of these subsets is given by the following product of multinomial coefficients.


  Suppose A=(a_i,j) is an integer M-admissible matrix with dimensions r× s. Then,
  
    |_A(M)| ∏_i=1^r ∑_j=1^s a_i,ja_i,1,a_i,2,…,a_i,s × ∏_j=1^s ∑_i=1^r a_i,ja_1,j,a_2,j,…,a_r,j .




The ordering of points (increasing or decreasing) within a particular cell is fixed by the corresponding entry of M.
However, for each row and for each column, the choice of which points occur in which cells can be made arbitrarily and independently.
The multinomial coefficient in the first product counts the number of ways in which the points of a gridded permutation can be distributed horizontally between the cells in row i.
Similarly, the term in the second product counts the number of ways in which the points can be distributed vertically between the cells in column j.


We use M-admissible matrices of weight one to record the proportion of points in each cell.
We call such matrices M-distribution matrices.
To avoid having to worry about rounding the number of points in each cell to an integer, we make use of Baranyai's Rounding Lemma <cit.>, as a consequence of which we have the following result.


  Suppose Γ=(γ_i,j) is an M-distribution matrix, and n is any positive integer.
  Then there exists an integer M-admissible matrix A=(a_i,j) of weight n such that, for each i,j, we have
  |a_i,j-nγ_i,j|<1.


In light of this, if Γ=(γ_i,j) is an M-distribution matrix and n a positive integer,
we use _Γ n(M) to denote _A(M), where A is some integer M-admissible matrix of weight n each of whose entries differs from the corresponding entry of nΓ by less than one, the existence of such an A being guaranteed by Proposition <ref>.
The exact choice of A is of no consequence to our arguments.
Note that in any gridded permutation in _Γ n(M) the proportion of points in cell (i,j) differs from γ_i,j by less than 1/n.





 §.§ Asymptotics of gridded permutations with a given distribution



Suppose τ=γ_1+γ_2+…+γ_k, where each γ_i>0.
Then Stirling's approximation gives the following asymptotic form for a multinomial coefficient.

    τnγ_1 n,γ_2 n,…,γ_k n ∼ √(τ/(2π)^k-1γ_1γ_2…γ_k)
    n^-(k-1)/2(τ^τ/γ_1^γ_1γ_2^γ_2…γ_k^γ_k)^n
    .


Hence, by Proposition <ref>, we have the following asymptotic enumeration of _Γ n(M).

If Γ=(γ_i,j) is an M-distribution matrix with
row sums ρ_i=∑_jγ_i,j
and
column sums κ_j=∑_iγ_i,j,
then

    |_Γ n(M)|  ∼  C  n^β g^n ,

where

    g  g(Γ) ∏_i ρ_i^ρ_i/∏_j γ_i,j^γ_i,j × ∏_j κ_j^κ_j/∏_i γ_i,j^γ_i,j
    ,

and C and β are constants that only depend on Γ.

Note that g(Γ)=|_Γ n(M)|^1/n is the exponential growth rate of those M-gridded permutations whose points are distributed between the cells in the proportions specified by Γ.

To avoid problems below, we take the products in the denominators of the expression for g(Γ) to be over nonzero entries of Γ only. 




 §.§ Maximising the growth rate


Given a grid class (M),
we would like to find an M-distribution matrix Γ=(γ_i,j) for which the growth rate g(Γ) is greatest.

We call such a matrix a maximal M-distribution matrix.
The following result specifies equations that must be satisfied by the entries of such a matrix.

7

  Suppose Γ=(γ_i,j) is a maximal M-distribution matrix.
  Then there exists a constant λ such that, for each nonzero entry γ_i,j of Γ, we have
  
    γ_i,j^2/ρ_iκ_jλ ,

  where ρ_i=∑_jγ_i,j and κ_j=∑_iγ_i,j are the row and columns sums of Γ.




We want to determine which choices of values for the γ_i,j maximise g(Γ), subject to the requirement that ∑γ_i,j=1.
This is a constrained optimisation problem, so we use the method of Lagrange multipliers.
To simplify the algebra, we maximise log g(Γ), which is of course equivalent to maximising g(Γ).
Viewing the nonzero γ_i,j as formal variables, we introduce the auxiliary function

    L(Γ,μ) log g(Γ)  - μ(1-∑γ_i,j) .


Then Γ must satisfy

    ∂/∂μ L(Γ,μ)  0
        and    ∂/∂γ_i,j L(Γ,μ)  0,

for each nonzero γ_i,j.
The first of these requirements is simply 
the constraint that ∑γ_i,j=1 (that is, Γ has weight one).

Now,

    log g(Γ) ∑_i (ρ_ilogρ_i - ∑_j γ_i,jlogγ_i,j)
     + ∑_j (κ_jlogκ_j - ∑_i γ_i,jlogγ_i,j)
    .

Given i and j, for brevity let γ=γ_i,j.
Also let ρ=ρ_i-γ and κ=κ_j-γ be the sum of the entries other than γ in row i and in column j of Γ, respectively.
Both ρ and κ are independent of γ, as are the contributions to L(Γ,μ) from any other row or column.
Then,

    ∂/∂γ_i,j L(Γ,μ) 
       ∂/∂γ( (ρ+γ)log(ρ+γ)  + (κ+γ)log(κ+γ)  -  2γlogγ + μγ) 
       ( 1 + log(ρ+γ) )  + ( 1 + log(κ+γ) )  -  2( 1 + logγ + μ) 
       logρ_i  + logκ_j  -  2logγ_i,j + μ .

Thus,

    ∂/∂γ_i,j L(Γ,μ)  0
      only if  γ_i,j^2/ρ_iκ_j e^μ .

The result follows by letting λ=e^μ.





 §.§ Uniqueness of the maximal distribution for connected classes


Albert and Vatter <cit.> give a different characterisation of maximal M-distribution matrices (in a more general context),
using the theory of singular value decompositions of matrices.

Given a real matrix A, a non-negative real number s is a singular value for A if there exist unit vectors 𝐮 and 𝐯 such that
A𝐯=s𝐮 and A^𝖳𝐮=s𝐯.
The vectors 𝐮 and 𝐯 are called left singular and right singular vectors for s, respectively.
If s is a singular value for A then s^2 is an eigenvalue of both AA^𝖳 and A^𝖳A.
Moreover, every left singular vector for s is an eigenvector of AA^𝖳 for the eigenvalue s^2,
and every right singular vector for s is an eigenvector of A^𝖳A for the eigenvalue s^2.



Given a gridding matrix M, let  be the binary 
matrix such that _i,j=0 if M_i,j is blank, and _i,j=1 otherwise.
Maximal M-distribution matrices can be characterised as follows.


  Suppose Γ=(γ_i,j) is a maximal M-distribution matrix.
  Then γ_i,j = _i,j 𝐯_i𝐮_j/s, where s is the greatest singular value of , and 𝐮 and 𝐯 are left and right singular vectors for s, respectively.


In general, left and right singular vectors may not be unique, so (M) may have multiple maximal M-distribution matrices.

However, if we limit which classes we consider, uniqueness can be guaranteed.

Given a gridding matrix M, its cell graph is the graph whose vertices are its non-blank cells,
and in which two vertices are adjacent if they share a row or a column and all the cells between them are blank.

For example, the cell graph of 41,-1,0,10,0,-10,-1,1 is
-6pt
[scale=0.25]
  3,3,0,3
  0,0,2
  0,1,1
  (1,3)–(4,3);
  (2,3)–(2,1)–(3,1)–(3,2);
.

If the cell graph of a gridding matrix is connected, then we also say that the matrix and the corresponding grid class are connected.

A connected grid class has a unique maximal distribution matrix.
To prove that this is the case, we extend the notion of connectivity to real matrices.
We say that a nonnegative matrix A is connected if there do not exist permutation matrices P and Q such that PAQ=R   0
0   S for some non-trivial R and S.


  If a nonnegative matrix A is connected, then its greatest singular value s 
  has unique left and right singular vectors 𝐮 and 𝐯, and
  the entries of 𝐮 and 𝐯 are all positive.


Consider A to be a weighted biadjacency matrix of a bipartite graph G.
If A is connected then so is G.
A A^𝖳 and A^𝖳A are then symmetric weighted adjacency matrices of the projections G_1 and G_2 of G onto its two vertex sets (in which nodes are adjacent if they share a neighbour in G). If G is connected, then so are G_1 and G_2. Thus A A^𝖳 and A^𝖳A are both irreducible.

 

  The result then follows from the Perron–Frobenius Theorem, given that s^2 is the principal eigenvalue of both AA^𝖳 and A^𝖳A,
  with 𝐮 and 𝐯 the corresponding (unique positive) principal eigenvectors.


With this, we can prove that connectivity is a sufficient condition for a grid class to have a unique maximal M-distribution matrix.



  If (M) is a connected grid class, then it has a unique maximal M-distribution matrix Γ=(γ_i,j).
  
  Moreover, γ_i,j is positive if and only if M_i,j is not blank.



  Since M is connected, then so is .
  Propositions <ref> and <ref> then guarantee that Γ is unique, with γ_i,j = _i,j 𝐯_i𝐮_j/s, where s is the greatest singular value of , and 𝐮 and 𝐯 are the unique left and right singular vectors for s.
  Since 𝐮 and 𝐯 are positive, γ_i,j>0
  if and only if M_i,j is not blank.


If M is connected, then we use Γ_M to denote the unique maximal M-distribution matrix.




 §.§ The distribution of points in a typical gridded permutation


We now prove that, if (M) is connected, then the distribution of points between cells in
almost all M-gridded permutations is close to that specified by Γ_M. 

Given any σ^#∈_n(M),
let σ^#_(i,j) denote the number of points of σ^# in cell (i,j), and
let Γ_σ^#=(σ^#_(i,j)/n) be the M-distribution matrix recording the proportion of the points of σ^# in each cell.



  If (M) is connected and
  
  Γ_M=(γ_i,j)
  
  is the unique maximal M-distribution matrix,
  then for any >0,
  
    max_i,j| σ^#_(i,j)/n - γ_i,j|  : σ^#∈_n(M) 1 .

  



  For brevity, let Γ=Γ_M,
  
  and let _n = {σ^#∈_n(M)
   : max_i,j| σ^#_(i,j)/n - γ_i,j| }
  consist of those M-gridded n-permutations whose distribution of points is -close to Γ,
  the proportion of points in each cell differing by no more than  from that specified by Γ.

  Also, let _n = _n(M) ∖_n be its complement,
  consisting of those M-gridded n-permutations whose distribution is -far from Γ,
  the proportion of points in some cell differing by more than  from that specified by Γ.

  For sufficiently large n, we have _Γ n(M) ⊆_n.
  Thus the exponential growth rate of _n is bounded below by g(Γ):
  
    (_n)   (_Γ n(M))  g(Γ) .


  Now let _n={Γ_σ^#:σ^#∈_n}
  be the collection of M-distribution matrices that record the distribution of points in M-gridded n-permutations whose distribution is -far from Γ.
  Then
  
  
  
  
    _n ⋃_Γ'∈_n_Γ'n(M) .

  Since M is connected, Γ is the only M-distribution matrix with maximal growth rate.
  Together with the continuity of the function g(·) and the compactness of the space of M-distribution matrices,
  this implies that
  there exists some g_<g(Γ) such that, for every Γ'∈⋃_n0_n, we have g(Γ')=(_Γ'n(M))  g_.

  Now |_n| < (n+1)^k, where k is the number of non-blank entries in M.
  Hence the exponential growth rate of _n is strictly bounded above by g(Γ):
  
    (_n)    g_ <  g(Γ) .

  Thus
  |_n|/|_n| = 0 as required.





 §.§ The shape of a typical permutation in a class


Theorem <ref> and Proposition <ref>
enable us to calculate the asymptotic distribution of points in almost all gridded permutations in any 
connected class.
From this we can also determine what the plot of a typical large 
permutation in such a class looks like.
See Figure <ref> for an illustration.



To formalise this idea,
we make use of certain probability measures on the unit square which act as analytic limits of sequences of permutations.

We briefly introduce only those parts of the theory that we need to state our result.
See the references below for more information.

A permuton is a probability measure μ on the unit square [0,1]^2 with uniform marginals:

μ ([a, b] × [0,1]) μ ([0,1] × [a, b])  b - a ,
  
  for every
  
0  a  b  1 .

A permuton can be associated with any permutation by taking its plot, scaling it into the unit square, and replacing the points with small squares,
as follows.

Suppose A⊂[0,1]^2 is a
rectangular region or a line segment.
Then we use
λ_A to denote the unique probability measure on [0,1]^2 with support A and mass uniformly distributed on A.
If σ is an n-permutation, then, for i=1,…,n, let S_σ(i) be the small square region [(i-1)/n,i/n]×[(σ(i)-1)/n,σ(i)/n].
Then
μ_σ = ∑_i=1^n 1/nλ_S_σ(i)
is the permuton corresponding to σ.
See Figure <ref> for an example.



Permutons were introduced in <cit.> with a different but equivalent definition.
The measure-theoretic perspective was first used (a little earlier) in <cit.>,
and was later applied in <cit.>, where the term “permuton” first occurs.

Given a class of permutations , it may be that almost all large permutations have the “same shape”.
Let _n be a random n-permutation drawn uniformly from _n.
If the sequence of random permutons (μ__n)_n1 converges in distribution to some permuton μ,
then we say that μ is the limit shape of .
A sequence of permutons (μ_n)_n1 converges in distribution to a permuton μ if,
for every (x,y)∈[0,1]^2,
we have μ_n([0,x]×[0,y]) = μ([0,x]×[0,y]).

If M 
is connected,
then Γ_M specifies the asymptotic distribution of points between the cells in a random permutation in (M).
Since any permuton has uniform marginals, this also specifies the asymptotic position of the cell dividers.
The points in each cell are monotonic,
and points in cells that share a row or column may be arbitrarily interleaved,
so the points in any cell asymptotically form a line segment across one of the diagonals of the cell.
Thus (M) has a deterministic limit shape μ_M defined as follows.

As in the formal definition of an M-gridding on page defMGridding,
for simplicity, here
we index matrices in the Euclidean manner from the lower left corner, counting rows from the bottom,
and with the column index before the row index.
Suppose Γ_M=(γ_i,j) is the unique maximal M-distribution matrix.

Let c_i=∑_i' i∑_jγ_i',j be the sum of the leftmost i columns of Γ_M,
and let r_j=∑_j' j∑_iγ_i,j' be the sum of the lowermost j columns of Γ_M.
Then
c_i gives the position of the ith column divider in μ_M, and
r_j gives the position of the jth row divider.
Let p_i,j = (c_i,r_j) be the point of their intersection.
Then the line segment in the support of μ_M for the non-blank cell (i,j) is given by

    L(i,j) {(1-x)p_i-1,j-1 +  x p_i,j :  x∈[0,1]},    if  M_i,j= , 
    {(1-x)p_i-1,j +  x p_i,j-1 :  x∈[0,1]},    if  M_i,j= .

Finally, the limit shape is constructed by distributing mass uniformly along the line segment for each non-blank cell:

    μ_M ∑_γ_i,j>0γ_i,jλ_L(i,j) .


Thus we have the following.


  If M 
  is connected,
  then μ_M, defined above, is the deterministic limit shape of (M).


For example, by applying Theorem <ref> and Proposition <ref>
we know that
the unique maximal 31,1,-10,-1,00,-1,0-distribution matrix is 1/6   1/3   1/6
0   1/6   0
0   1/6   0,
yielding the limit shape at the right of Figure <ref>.

It seems likely that fluctuations about the limit shape of a grid class can be described by coupled Brownian motions, as in the case of square permutations (permutations in which every point is a record) <cit.>.
However, that is beyond our scope here.
It should also be possible to determine the 
(possibly random) limit shapes of classes with more than one 
component.
See <cit.> for other recent work on the limit shapes of permutation classes.






§ CONNECTED CLASSES WITH ONE CORNER






We say that a cell C of a gridding matrix is a corner cell or just a corner, if there is both another non-blank cell in the same row as C and also another non-blank cell in the same column as C.
A cell that is not a corner is a non-corner cell.

For example, 41,-1,0,10,0,-10,-1,1 has three corners, one in the top row and two in the bottom row.

In this section we consider connected monotone grid classes that have a single corner.
These are either -shaped, -shaped or -shaped.
See Figure <ref> for an example of each.
-shaped and -shaped classes come in four different orientations.



The row containing the corner we call the main row, and
the column containing the corner we call the main column.
For brevity, non-corner cells in the main row are simply called row cells,
and non-corner cells in the main column are called column cells.
Thus each ,  or -shaped class consists of a corner cell, some row cells and some column cells.

Throughout this section
we assume
that (M) is a connected one-corner class with dimensions (r+1)×(c+1).
Thus M has
r+c+1 non-blank cells: the corner,
r column cells, and
c row cells.
For example, for the -shaped and -shaped classes in Figure <ref>, we have r=3 and c=4.

Our analysis of connected one-corner classes is as follows.
First, we apply Theorem <ref> to determine the asymptotic distribution of points between the cells (Section <ref>).
Then we use generating functions to establish the asymptotics of the number of gridded permutations in such a class (Section <ref>).
This is followed by an investigation of the ways in which points can dance between cells,
including the introduction of an appropriate notion for these classes of a constrained gridded permutation (Section <ref>).

We then introduce the different corner types that can appear in ,  and -shaped classes,
with Theorem <ref> giving the asymptotics for these classes in terms of the corner type.
This is then followed by a detailed examination of each of the corner types

(Sections <ref> to <ref>).
To conclude, we prove that almost all gridded permutations are constrained (Section <ref>).
Finally, we briefly consider how our approach can be extended beyond connected one-corner classes (Section <ref>).




 §.§ The distribution of points between cells


For a connected one-corner class, the asymptotic distribution of points between the cells in a typical M-gridded permutation can easily be determined from Theorem <ref> and Proposition <ref>.
The asymptotic proportion of points in each row cell satisfies the same equations and hence these proportions are all equal.
The same is true for the column cells.

We use α to denote the proportion of points in the corner cell,
β to denote the proportion in each of the row cells, and
γ the proportion in each of the column cells.
So, for example,
the unique maximal distribution matrix for the
-shaped class in Figure <ref> has the form











    γ         
       γ         
    β   α   β   β   β
       γ         .

Then, by Theorem <ref> and Proposition <ref>, we know that α, β and γ are the unique positive solutions to the equations

    α+cβ+rγ=1
      and  α^2/(α+cβ)(α+rγ) = β/α+cβ = γ/α+rγ
    .

Solving these then yields

    α1/q ,
        βc-r+q-1/2 c q ,
        γr-c+q-1/2 r q
    ,

where

    q √((r+c+1)^2-4 c r) .

Note that

    λβ/α+cβγ/α+rγα^2/(α+cβ)(α+rγ)r+c+1-q/2 r c

is the common value of the ratios from Proposition <ref>.

For example, if r=3 and c=4 then we have q=4, α=1/4, β=1/8, γ=1/12 and λ=1/6.





 §.§ The asymptotics of gridded classes


For skinny monotone grid classes, exact enumeration of the gridded permutations is simple:
a k-cell skinny grid class has exactly k^n gridded n-permutations (Proposition <ref>).
For non-skinny classes things are not so straightforward.

Our approach is to determine the generating function for the gridded class, using a technique first described in <cit.>.
We  then extract the asymptotic growth of the number of gridded permutations from the generating function using standard methods from analytic combinatorics.

To determine the generating function for the gridded permutations in an -shaped, -shaped or -shaped class
we “stitch together” two skinny classes, one formed by the main row and the other by the main column.

The bivariate generating function for the (c+1)-cell horizontal skinny gridded class ^# formed from the main row,
in which x is used to mark the points in the corner cell, is

    H^#(z,x) 1/1-cz-zx .

Similarly, the bivariate generating function for the (r+1)-cell vertical skinny gridded class ^# formed from the main column,
in which y is used to mark the points in the corner cell, is

    V^#(z,y) 1/1-rz-zy .

Thus, the set of pairs (σ_𝗁^#, σ_𝗏^#), consisting of
an -gridded permutation σ_𝗁^#
and
a -gridded permutation σ_𝗏^#,
is enumerated by the product of the generating functions of the two skinny classes:

    P^#(z,x,y)  = 
    H^#(z,x)V^#(z,y)  = 1/(1-cz-zx)(1-rz-zy) .


To count M-gridded permutations, we are only interested in those pairs (σ_𝗁^#, σ_𝗏^#)
for which
the number of points of σ_𝗁^# in the corner cell is the same as
the number of points of σ_𝗏^# in the corner cell.
Any M-gridded permutation can be decomposed into such a pair,
and any such pair combine in a unique way to form an M-gridded permutation, as illustrated in Figure <ref>.



In terms of the generating function P^#(z,x,y),
we need to extract the terms in which x and y have the same exponent.
However, we also need to correct for the double-counting of the points in the corner cell.
Thus we want

    ∑_m0[x^m y^m]P^#(z, x/√(z), y/√(z)) .

Here, the division of the second and third arguments by √(z) decreases the exponent of z by one for each point in the corner.

To extract the terms in which x and y have the same exponent, we let y=x^-1.
This yields a Laurent series[A Laurent series is a power series in which terms of negative degree are permitted.] in x, where x now records for each pair (σ_𝗁^#, σ_𝗏^#) the difference between the number of points of σ_𝗁^# in the
corner
and the number of points of σ_𝗏^# in the
corner.
We just want the constant term (when this difference is zero):

    [x^0]P^#(z, x/√(z), x^-1/√(z)) .


4
To extract this constant term, we use the following result.

  If f(x)=f(z,x) is a Laurent series, then the constant term [x^0]f(x) is given by the sum of the residues[The residue of h(x) at x=α is 
  the coefficient of (x-α)^-1 in the Laurent expansion of h(x) around x=α. If α is a simple pole, then this is just the value of yh(y+α) at y=0.] of x^-1f(x) at those 
  poles α(z) of f(x) for which lim_z→0α(z)=0. These are known as the small poles.


In our case,

    x^-1P^#(z, x/√(z), x^-1/√(z)) 1/(1-cz-x√(z))(x-rzx-√(z))

has two poles, one at x_1(z)=(1-cz)/√(z), and the other at x_2(z)=√(z)/(1-rz).
Only x_2(z) is small.
Thus all we need is the residue at x=x_2(z).
This then yields the following generating function for M-gridded permutations in a connected class with one corner:

    F_M^#(z)
    ∑_n 0|_n(M)|   z^n
    1/1-(r+c+1)z + rcz^2 .


We  extract the asymptotic growth of the number of gridded permutations from this generating function by using the following standard result.

  Suppose F(z) is the ordinary generating function of a combinatorial class .
  Let ρ be the least singularity of F(z) on the positive real axis.
  If there are no other singularities on the radius of convergence and ρ is a pole of order r, then
  
    |_n|  ∼  cρ^-nn^r-1    where  
      c  = ρ^-r(r-1)!lim_z→ρ (ρ-z)^rF(z) .


In the case that F(z) is a rational function with a denominator Q(z) of degree d, the exponential growth rate ρ^-1 is the greatest root of the polynomial z^dQ(z^-1).
Thus the growth rate of ^#(M), and hence also of (M), equals the larger of the two roots of the quadratic equation z^2 - (r+c+1)z+rc=0:

    g_M ((M)) 12 (r+c+1+q),

where q is defined in equation (<ref>) on page eqQ.
From <cit.>, we know that g_M=g(Γ_M), so g_M could also have been calculated from the proportions in equation (<ref>) by using Proposition <ref>.

Since z=g_M^-1 is a simple pole of F_M^#(z), the subexponential term is just a constant:

    θ_M^#(n)  (r+c+1+q)/2q  g_M/q .

Thus the asymptotic growth of the number of gridded permutations in a connected one-corner class is given by the following proposition.

  If M is connected with one corner and has dimensions (r+1)×(c+1), then

    |_n(M)|  ∼ θ^#g^n,
      where  θ^#r+c+q+1/2q  and  
    g r+c+q+1/2 ,

with q √((r+c+1)^2-4 c r).

For example, if M has dimensions 4×5 (r=3 and c=4), then |_n(M)|∼3/2×6^n.




 §.§ Dancing: peaks, diagonals and tees


Generalising from skinny classes, we say that two non-blank cells that are adjacent (either horizontally or vertically) in a gridding matrix form a peak if one is increasing () and the other is decreasing ().
Thus, in addition to peaks that point up or down, we also have peaks that point left (11-1) and peaks that point right (1-11).

Given a gridded permutation σ^# with at least two points in each cell in a gridded class with a peak Λ that points left or right, the peak point of Λ is the leftmost of the points of σ^# in the two cells of Λ if Λ points left, and is the rightmost of the points of σ^# in the two cells of Λ if Λ points right.
For example, the rightmost circled point in the gridded permutation at the right of Figure <ref> on page figHVStitching is the peak point of a peak that points right.

Generalising the notion of dancing we introduced in Section <ref>,
if Q is a peak point 

adjacent to the row or column divider which separates the two cells of the peak,
then Q can dance.
The movement of this divider to the other side of Q results in another valid gridding.

For one-corner classes, in addition to peaks, we also need to take into account diagonally adjacent pairs of cells that form diagonals, either  or .
For example, the -shaped class at the right of Figure <ref> on page figLTX has two diagonals. 

In addition, two peaks may combine to form a tee: that is, any of the four rotations of .
For example, the -shaped class in the centre of Figure <ref> has a tee. 

We say that a peak is a corner peak if one of its two cells is a corner cell,
and that it is a non-corner peak otherwise.
For example, the -shaped class at the left of Figure <ref> has one corner peak and four non-corner peaks.
In connected one-corner classes, 
diagonals and tees only occur at corners:
diagonals can only be formed by two cells which are adjacent to a corner,
and tees can only be formed by a corner cell and two of its neighbours.

The presence of a diagonal or tee makes possible new ways for points to dance.
The dancing of a peak point, which we now call peak dancing or dancing at a peak, involves only a single point.
In contrast, diagonal dancing and tee dancing may involve more than one point.



Suppose that M is a gridding matrix, with a diagonal formed from two cells
separated by a row divider and a column divider.
Suppose that σ^# is an M-gridded permutation, and that Q is a point of σ^#
in
one of the two cells of
the diagonal.
If the simultaneous movement of both dividers immediately to the other
side of Q results in another valid M-gridding of σ,
then we say that Q can dance diagonally.
See Figure <ref> for an illustration.
Note that any points that lie both horizontally and vertically between Q and the intersection of the dividers can also dance.
This yields a monotone sequence of points all of which are able to dance.

Similarly, suppose that M is a gridding matrix, with a tee formed from a corner cell and two other cells, C_1 and C_2 say,
separated by a row divider and a column divider.
Suppose that σ^# is an M-gridded permutation, and that Q is a point of σ^#
in
one of the three cells of
the tee.
If there is an alternating sequence of one-step moves of the two dividers
which leaves one of them immediately the other side of Q
and results in another valid M-gridding of σ,
then we say that Q can dance through the tee.
See Figure <ref> for an illustration.




Again, any points that lie between Q and the intersection of the dividers can also dance,
yielding a monotone sequence of points that can dance.
Note, however, that the first (or last) point in this sequence, lying in C_1 say, may only be able to dance into the corner, and not be able to dance through to C_2.
For example, the rightmost circled point in the gridded permutations in Figure <ref> can't dance into the cell below the corner.

For almost all permutations in the classes we consider,
the valid griddings are restricted to those that can be obtained through
peak dancing, diagonal dancing and tee dancing.
So, generalising the definition for skinny classes, if (M) is a connected one-corner class
we say that an M-gridded permutation σ^# is M-constrained (or just constrained) if

  
  (a) every M-gridding of its underlying permutation σ is the result of zero or more points of σ^# dancing at a peak or diagonally or through a tee,
  and
  
  (b) in every M-gridding of σ, each non-blank cell contains at least two points.

We defer further analysis and a proof that most M-gridded permutations are constrained until Section <ref>, after a discussion of the different corner types.





 §.§ Counting griddings


To complete our computation of the asymptotic growth of

connected one-corner
classes,
we determine how a constrained M-gridded permutation σ^# must be structured so that its underlying permutation σ has a given number of griddings.
This depends on the orientation (either increasing or decreasing) of the corner cell and the orientation of each of its non-blank neighbours.
Taking symmetries into account, there are eleven inequivalent corner types to consider.
These are shown in Table <ref> on page tblKappa below.

By combining this structural analysis with the asymptotic distribution of points between cells from Section <ref>,
we then calculate, for each ℓ1, the asymptotic probability

    P_ℓσ has exactly ℓ distinct M-griddings : σ^#∈_n(M) .

Then, letting

    κ_M
    ∑_ℓ1P_ℓ/ℓ
    |_n(M)|/|_n(M)|

be the correction factor for the class,
we conclude that

|_n(M)|
∼κ_M  θ^#  g^n ,

where θ^# and g are given by Proposition <ref>.

To determine the probabilities, we make repeated use of the following observation, recalling the definition of the common ratio λ in equation (<ref>) on page eqLambda.

  Suppose (M) is a connected one-corner class with dimensions (r+1)×(c+1), and
  let α, β and γ be the asymptotic proportion of points of an M-gridded permutation
  in the corner cell, in any row cell, and in any column cell, respectively.

  Then, in an M-gridded permutation, for each k1, the asymptotic probability that the kth point from the top (or bottom) in the main row occurs in any given row cell equals λ=β/(α+cβ).
  Similarly, the asymptotic probability that the kth point from the left (or right) in the main column occurs in any given column cell also equals λ=γ/(α+rγ).

  Moreover, for j≠ k, the events that the jth and kth points from the top or bottom 
  in the main row 
  occur in specific cells are asymptotically independent.
  And analogously for points in the main column.


Note also that if dancing can occur asymptotically independently in more than one location, then the corresponding correction factors multiply.
This follows from the following arithmetic observation.

  Given real sequences (P'_i)_i1 and (P”_j)_j1,
  let κ'=∑_i1P'_i/i and κ”=∑_j1P”_j/j.
  Suppose P_ℓ =∑_ij=ℓP'_i P”_j.
  Then,
  
    ∑_ℓ1P_ℓ /ℓκ' κ” .



  
    κ' κ”∑_i1P'_i/i ∑_j1P”_j/j∑_i,j1P'_i P”_j/ij∑_ℓ11/ℓ∑_ij=ℓ P'_i P”_j
      ∑_ℓ1P_ℓ/ℓ.



4
Non-corner peaks


Before looking at each corner type in turn, we consider the effect of non-corner peaks.
By definition, in any constrained M-gridded permutation,
at each non-corner peak of M
there is a peak point which can dance.
So, if M has p non-corner peaks, these contribute a factor of 2^p to the number of possible M-griddings, in an analogous manner to skinny classes (Proposition <ref>).
Thus, since almost all M-gridded permutations are M-constrained
(see Theorem <ref> below),
the asymptotic enumeration of ,  and -shaped classes is given by the following result.


  Suppose (M) is a connected one-corner class with corner type τ, and p non-corner peaks, then
  
    |_n(M)|
       ∼ 
      2^-p κ(τ)  θ^# 
      g^n ,

where κ(τ) is the correction factor for a gridding matrix with the same corner type and dimensions as M but with no non-corner peaks,
and θ^# and g are as given by Proposition <ref>.


Note that the correction factor for the class is given by κ_M=2^-p κ(τ).

4
Worked example


Before turning to the calculation of the correction factors for each corner type, we very briefly illustrate our method 
with an example, by determining the asymptotic enumeration of the 
 class from Figure <ref>:

    M_𝖷50,-10,11,1,1,-1,10,1.












Firstly, from Proposition <ref>, since r=3 and c=4, we have |_n(M_𝖷)|∼3/2×6^n.

Secondly, M_𝖷 has three non-corner peaks and has corner type X_0. 
Now, κ(X_0)=(1-λ)^2, and from equation (<ref>) we know that λ =1/6.

Thus, by Theorem <ref>, we have

    |_n(M_𝖷)|  ∼  2^-3×(56)^2 ×32 × 6^n 25192× 6^n.


4
Correction factors for corner types


We now calculate
κ(τ) for each 
corner type τ.

Table <ref> summarises our results.



The subscripts in the names used for corner types are determined by reading the non-blank cells in normal reading order and treating  as 0 and  as 1 to give a binary number.
For example, T_5 is 3-1,1,-10,1 since 5=0101_2.
The symmetry giving the least subscript is chosen as the representative.
Note that rotating a class by 90^∘ switches the roles of r and c and of β and γ. We use τ to denote a 90^∘ rotation of corner type τ.

In the figures used below for each corner type, dots indicate where additional row and column cells may occur, as long as they don't create additional corner peaks, diagonals or tees.

We begin with the simplest of the corner types.

7


  §.§.§ Corner type L0




.12522002-1,-1-1 

This corner type has no peaks, diagonals or tees.
It can't occur in a -shaped or -shaped class.
In a constrained gridded permutation, no dancing is possible, so the underlying permutation has a single gridding.
Thus P_1=1, and P_ℓ=0 if ℓ>1.
Hence κ(L_0)=1.


10


 §.§ Corners with peaks


7


  §.§.§ Corner type L1




.125
22002-1,-11
-22102-1,-10,1
-2232103-1,-1,-10,1
3.67pt-22101520,-1-1,-10,1

This corner type has one peak.
It can't occur in an -shaped class.
As can be seen from the figures, the peak may be orientated in different ways with respect to the other cells adjacent to the corner.
However, the analysis is the same in every case, so we consider this to be a single corner type.

Given a constrained gridded permutation,
let Q be the peak point, and let R be the lowest point in any of the row cells.
Q is the only point that may be able to dance.
It can't dance if it is above R in the corner cell.
Otherwise (if it is below R, either in the corner cell or in the cell below the corner) it can dance.
We say that point R is the controller, since it controls whether Q can dance or not.
See Figure <ref> for an illustration of the three cases.



The peak point, Q, is either in the corner or in the cell immediately below the corner.
Thus the asymptotic probability that Q is in the corner equals α/(α+γ),
and the asymptotic probability that the lowest point in the main row is not in the corner equals cβ/(α+cβ).
These events are asymptotically independent, so

    Q can't dance ∼ 
    P_1 α/α+γ×cβ/α+cβcαλ/α+γ    and    
    P_2  1-P_1 .

Hence,

    κ(L_1)
      P_1 + 12 P_2
    1/2(1+c αλ/α + γ)
    2 r/3 r-c+q-1 .


7


  §.§.§ Corner type L4




.125
220021,-1-1
-22321031,1,-10,-1 


This corner type has two peaks.
It can't occur in an -shaped class.
Given a constrained gridded permutation,
let Q_1 be the peak point at the left and Q_2 be the peak point at the top.
The same analysis as for L_1 gives

    Q_1 can't dance ∼ cαλ/α+γ,
        Q_2 can't dance ∼ rαλ/α+β.

The event that Q_1 can dance depends on the points adjacent to the row divider below the corner,
whereas the event that Q_2 can dance depends on the points adjacent to the column divider to the right of the corner.
Thus these events
are asymptotically independent.
So, by Observation <ref>, we have

    κ(L_4)
    κ(L_1)  κ(L_1)
    1/4(1+r αλ/α + β) (1+c αλ/α + γ)
    4rc/(3r-c+q-1)(3c-r+q-1) .


9


  §.§.§ Corner type T5




.125
-2232103-1,1,-10,1
3.67pt-2232101530,1-1,1,-10,1 


This corner type also has two peaks.
It can't occur in an -shaped class.
Given a constrained gridded permutation,
let Q_1 and Q_2 be the two peak points.
The same analysis as for L_1 gives

    Q_1 can't dance ∼ Q_2 can't dance ∼ rαλ/α+β.

The event that one of the peak points can dance depends on the points adjacent to the column divider to the left of the corner,
whereas the event that the other peak point can dance depends on the points adjacent to the column divider to the right of the corner.
Thus these events
are asymptotically independent.
So, by
Observation <ref>, we have

    κ(T_5)
    κ(L_1)^2
    1/4(1+r αλ/α + β)^24c^2/(3c-r+q-1)^2 .



10


 §.§ Corners with diagonals


7


  §.§.§ Corner type L7




.125
220021,11
-22321031,1,10,1 


This corner type has one diagonal.
It can't occur in an -shaped class.
Let  be the cell immediately to the right of the corner, and  be the cell immediately below the corner.



Given a constrained gridded permutation,
let Q_1 be the lowest point in the main row that is not in .
Note that Q_1 may be in the corner cell.
Let k_1 be the number of points in  that lie below Q_1.
Similarly, let Q_2 be the rightmost point in the main column that is not in  (Q_2 may also be in the corner cell),
and let k_2 be the number of points in  that lie to the right of Q_2.
These k_1+k_2 points can dance diagonally, giving k_1+k_2+1 distinct griddings of the underlying permutation.
Note that both k_1 and k_2 may be zero.
Points above Q_1 and points to the left of Q_2 can't dance.
Points Q_1 and Q_2 are the controllers, because they control which points can dance.
See Figure <ref> for an illustration.

Now, by Observation <ref>,
for each i0, we have

    k_1=i ∼ λ^i(1-λ)      and also     k_2=i ∼ λ^i(1-λ) .

The value of k_1 is asymptotically independent of the value of k_2.
Thus, for each ℓ1, the asymptotic probability of having exactly ℓ griddings is

    k_1+k_2+1=ℓ∑_k_1=0^ℓ-1k_1=ik_2=ℓ-1-i ∼ ℓ λ^ℓ-1(1-λ)^2 .

Hence,

    κ(L_7) ∑_ℓ=1^∞λ^ℓ-1(1-λ)^2
     1-λ 1- r+c+1-q/2 r c.


7


  §.§.§ Corner type T2




.125
-2232103-1,-1,10,-1
3.67pt-2232101530,-1-1,-1,10,-1 


This corner type has one peak and one diagonal.
It can't occur in an -shaped class.

Dancing at the peak depends on the points adjacent to the column divider to the right of the corner,
whereas dancing on the diagonal
depends on the points adjacent to the column divider to the left of the corner and
points adjacent to the row divider below the corner.
So these
are asymptotically independent.
Thus, from the analysis for L_1 and L_7, and by Observation <ref>, we have

    κ(T_2)
    κ(L_1)  κ(L_7)
    1/2(1+r αλ/α + β) (1-λ)
    .


7


  §.§.§ Corner type X0




.125-2232101530,-1-1,-1,-10,-1 

This corner type has two diagonals.
It can't occur in an -shaped or -shaped class.

Dancing on one of the diagonals depends on the points adjacent to the column divider
to the left of the corner and points adjacent to the row divider below the corner,
whereas
dancing on the other diagonal depends on the points adjacent to the column divider
to the right of the corner and points adjacent to the row divider above the corner.
So these
are asymptotically independent.
Thus, from the analysis for L_7, and by Observation <ref>, we have

    κ(X_0)
    κ(L_7)^2
     (1-λ)^2
    .



7


  §.§.§ Corner type X7




.125-2232101530,-1-1,1,10,1 

This corner type has two peaks and one diagonal.
It can't occur in an -shaped or -shaped class.

Dancing at the two peaks depends on the points adjacent to the column divider
to the left of the corner and points adjacent to the row divider above the corner,
whereas
dancing on the diagonal depends on the points adjacent to the column divider
to the right of the corner and points adjacent to the row divider below the corner.
So these
are asymptotically independent.
Thus, from the analysis for L_4 and L_7, and by Observation <ref>, we have

    κ(X_7)
    κ(L_4) κ(L_7)
    1/4(1+r αλ/α + β) (1+c αλ/α + γ) (1-λ)
    .


10



 §.§ Corners with tees


7


  §.§.§ Corner type L3




.125
22002-1,11
-2232103-1,-1,10,1
3.67pt-2232101530,-1-1,-1,10,1 


This corner type has one tee.
It is the only one of the eleven corner types that can occur in -shaped, -shaped and -shaped classes.

Analysis of this corner type is considerably more complex than for the corners we've considered so far.
We analyse the main row by reading its points from the bottom.
Similarly, we analyse the main column by reading its points from the right.
Let C denote the corner cell,
let  be the cell immediately to the right of C,
and let  be the cell immediately below C.

4
Sequences of points

We represent the sequence of points in the main row, when read from the bottom, by words over the alphabet {,,}.
The letter  represents a point in the corner cell C,
the letter  represents a point in ,
and the letter  represents a point in some other cell of the main row.
We use subscripts to identify specific points.
Thus _Q represents the occurrence of point Q in the corner cell.
For example, the sequence of points in the main row of the gridded permutation at the left of Figure <ref> 
is represented by the word
_Q  _R   _S.

Simple regular expressions are used to denote sets specifying the possible ordering of the initial points.
In these regular expressions, a^ denotes a sequence of zero or more copies of letter a.
For example, ^_Q consists of arrangements of points in the main row in which Q is the lowest point in the corner cell (_Q being the first occurrence of ) and where Q is below any points in , the only points below Q occurring elsewhere (represented by ^).
Note that any ordering of points is permitted after the initial points specified by the regular expression.

By Observation <ref>, associated with ,  and  we have the following asymptotic probabilities:




    ,      ,      .

By symmetry, for the location of points in the main column, when read from the right, the same approach yields the following asymptotic probabilities:

    1-rλ,      λ,       (r-1)λ.


The asymptotic probability that the points in the main row are in (the set denoted by) some regular expression is given by the product formed by replacing each  by , each  by  and each  by , and replacing each ^, ^ and ^ by 1/(1-), 1/(1-) and 1/(1-), respectively.
For example, the asymptotic probability that the arrangement of points in the main row is in ^_Q equals /(1-).

4
The number of griddings

Recall, from the introduction to tee dancing on page defDiagDancing, that a monotone sequence of points can dance.
The points not at one of the ends of the sequence can dance between  and  through C.
However, the first and last of these points may only be able to dance between  and C or between  and C, but not from  to .

Each point that can dance between  and  contributes two to the number of 
griddings.
However, a point that can only dance between C and either  or  contributes just one additional gridding.
In our analysis below, we determine the contribution to the number of griddings from points in the main row, which we denote m_1,
and also the contribution to the number of griddings from points in the main column, which we denote m_2.
Including the original gridding, the total number of distinct griddings is then m_1+m_2+1.

4
Labelled points

Given a constrained gridded permutation,
let Q be the lowest point in the corner cell C.
Our analysis depends on whether or not Q is the peak point of each of the two peaks that form the tee.

If c>1 (so we have more than two columns), then let S be the lowest point in the main row that is in a cell other than C or .
Similarly, if r>1, then let T be the rightmost point in the main column that is in a cell other than C or .

We consider three cases.
In the first, Q is the peak point of both of the peaks that form the tee.
In the second, Q is not the peak point of either of the peaks that form the tee.
Finally, in the third, Q is the peak point of just one of the two peaks. 



  §.§.§ Case 1: Q is the peak point of both of the peaks that form the tee


In Case 1, point Q is lower than every point in  and to the right of every point in .
If S is above Q, then Q is adjacent to the row divider below C and can dance vertically into , contributing one to the number of griddings.
Similarly, if T is to the left of Q, then Q is adjacent to the column divider to the right of C and can dance horizontally into , again contributing one to the number of griddings.

Let R be the second lowest point in the corner cell C.
That is, R is the point immediately above Q in the corner cell.
Points R, S  and T are the controllers, because they control which points can dance.

Let k_1 be the number of points in  below both R and S (if c>1).
These k_1 points are above Q, and thus can all dance through the tee to , each contributing 2 to the number of griddings.
Similarly, let k_2 be the number of points in  to the right of both R and T (if r>1).
These k_2 points are to the right of Q, and can also all dance through to , again each contributing 2 to the number of griddings.
Note that k_1 and k_2 may be zero.
See Figure <ref> for two gridded permutations satisfying the conditions of Case 1.



In Case 1, the possible ordering of the points in the main row, reading from the bottom, is given by the regular expression ^_Q, the only points  below Q (if any) being in cells other than C and .
So, the asymptotic probability that the points in the main row of a gridded permutation satisfy the conditions of Case 1 is given by
    
    p_1Case 1/1-/1-
        .


For both the main row and the main column, we have three subcases.
We analyse the main row, reading its points from the bottom.
The analysis of the main column is analogous (reading its points from the right).
The three subcases are as follows:


    5
    
  1a. 
    S, if it exists, is above Q, which is the lowest point in the main row.
    S (if it exists) may be above or below R.
    There is no point in  that is above S and below R.
    See the left of Figure <ref> for examples in both the main row and the main column.

    Q can dance vertically into . Thus m_1=2k_1+1.

    The possible ordering of the points in the main row, reading from the bottom, is given by the regular expression _Q ^^_R.
   
    So, for each i0, we have
    
    p_1a(i) 
        Case 1a and k_1=i^2  ^i/1-()^2  ^i/1-
        .


    5
    
  1b. 
    S is above Q and below R, and at least one point in  is above S and below R.
    Again, Q is the lowest point in the main row.
    Let U be the lowest of these points.
    Note that U is not one of the k_1 points in  below both R and S.
    Other points not in C or  may lie below U.
    See the right of Figure <ref> for an example.

    Q can dance vertically into , contributing one to the number of griddings.
    And U can dance horizontally into the corner cell, but not from the corner into , so this contributes another gridding.
    Thus m_1=2k_1+2.

    The possible ordering of the points in the main row, reading from the bottom, is given by the regular expression _Q ^_S ^_U.
    So, for each i0, we have
    
    p_1b(i) 
        Case 1b and k_1=i ^i+1 /1-()  ^i+1 /1-
        .


    
  1c.
    S is below Q.
    So Q can't dance vertically into , and k_1=0, since every point in  is above Q and hence also above S.
    Thus m_1=0.
    See the right of Figure <ref> for an example of this in the main column,
    in which the controller T is to the right of Q, so Q can't dance horizontally into , and m_2=k_2=0.

    The possible ordering of the points in the main row, reading from the bottom, is given by the regular expression _S ^_Q.
    Thus,
    
    p_1c
        Case 1c and k_1=0 /1-()  /1-
        .




Table <ref> gives the total number of griddings for each possibility in Case 1.



5
We now the calculate, for each ℓ1, the probability P_1(ℓ) := Case 1 and ℓ griddings.

Let q_1a(i), q_1b(i) and q_1c be the subcase probabilities for the main column,
formed from p_1a(i), p_1b(i) and p_1c
by replacing , ,  with
, , ,
respectively.

Then, from Table <ref>, we have

    P_1(1)     p_1c  q_1c , 
    
    P_1(2)     p_1a(0)   q_1c +  p_1c  q_1a(0) , 
    
    P_1(3)     p_1a(0)   q_1a(0)  +  p_1b(0)   q_1c +  p_1c  q_1b(0) , 
    
    P_1(ℓ)     +  p_1a(ℓ-22)   q_1c +  p_1c  q_1a(ℓ-22) 
        + ∑_i=0^(ℓ-4)/2( p_1a(i)   q_1b(ℓ-42-i)
                            +  p_1b(ℓ-42-i)   q_1a(i) )
                           ,   ℓ4, even , 
    
    P_1(ℓ)     +  p_1b(ℓ-32)   q_1c + p_1c  q_1b(ℓ-32) 
        + ∑_i=0^(ℓ-3)/2 p_1a(i)   q_1a(ℓ-32-i)
                                            + ∑_i=0^(ℓ-5)/2 p_1b(i)   q_1b(ℓ-52-i)
                           ,   ℓ5, odd .


After simplification, facilitated by using a computer algebra system, this yields

    P_1(ℓ) λ^(ℓ-3)/2   ((ℓ-1)   + (ℓ+1) λ  ) 2  (1-)  (1-) ,    ℓ1, odd , 
    ℓ λ^(ℓ-2)/2    (  +  )2  (1-)  (1-) ,    ℓ2, even .




  §.§.§ Case 2: Q is not the peak point of either of the peaks that form the tee


In Case 2, there is a point in  below Q and a point in  to the right of Q.
So Q can't dance into either  or .
Points Q, S  and T are the controllers, controlling which points can dance.

Let k_1 be the number of points in  below both Q and S (if c>1).
These k_1 points can all dance through the tee to , each contributing 2 to the number of griddings.
Similarly, let k_2 be the number of points in  to the right of both Q and T (if r>1).
These k_2 points can also all dance through to , again each contributing 2 to the number of griddings.
See Figure <ref> for a gridded permutation satisfying the conditions of Case 2.
Again, k_1 and k_2 may be zero.




For both the main row and the main column, we have two subcases.
We analyse the main row, the analysis of the main column being analogous.


    
  2a. 
    There is no point in  above S and below Q.
    S may be above or below Q, or S may not exist.
    Since there is a point in  below Q (by the definition of Case 2), in this subcase (only) we know that k_1>0.

    The k_1 points in  below both Q and S (if it exists) can all dance through to .
    Thus m_1=2k_1.

    The possible ordering of the points in the main row, reading from the bottom, is given by the regular expression ^^_Q.
    So, for each i1, we have
    
    p_2a(i) 
        Case 2a and k_1=i ^i/1-()  ^i/1-
        .


    
  2b. 
      S is below Q with at least one point in  above S and below Q.
      Let U be the lowest of these points.
      Note that U is not one of the k_1 points in  that are below both Q and S.
      Other points not in C or  may lie below U.

      U can dance horizontally into the corner cell, but not from the corner into .
      Thus m_1=2k_1+1.

    The possible ordering of the points in the main row, reading from the bottom, is given by the regular expression ^_S ^_U.
    So, for each i0, we have
    
    p_2b(i) 
        Case 2b and k_1=i^i+1 /1-^i+1 /1-
        .



Table <ref> gives the total number of griddings for each possibility in Case 2.




We now calculate, for each ℓ1, the probability P_2(ℓ) = Case 2 and ℓ griddings.


Let q_2a(i) and q_2b(i) be the subcase probabilities for the main column,
formed from p_2a(i) and p_2b(i)
by replacing , ,  with
, , ,
respectively.

Then, from Table <ref>, we have

    P_2(3)     p_2b(0)   q_2b(0) , 
    
    P_2(ℓ)    ∑_i=1^(ℓ-2)/2( p_2a(i)   q_2b(ℓ-22-i)
                             +  p_2b(ℓ-22-i)   q_2a(i) )
                            ,   ℓ4, even , 
    
    P_2(ℓ)    ∑_i=1^(ℓ-1)/2-1 p_2a(i)   q_2a(ℓ-12-i)
                            + ∑_i=0^(ℓ-3)/2 p_2b(i)   q_2b(ℓ-32-i)
                           ,   ℓ5, odd .


After simplification, this yields

    P_2(ℓ) 
    0 ,    ℓ=1 , 
    (ℓ-2) λ^ℓ/2  (  +  )2  (1-)  (1-) ,    ℓ2, even , 
    λ^(ℓ-1)/2 ((ℓ-3)   + (ℓ-1) λ  ) 2  (1-)  (1-) ,    ℓ3, odd .




  §.§.§ Case 3: Q is the peak point of just one of the two peaks that form the tee


Case 3 combines Case 1 for the main row and Case 2 for the main column, or vice versa.

Suppose the main row satisfies Case 1 and the main column satisfies Case 2, as in Figure <ref>.
Then Q can't dance into either  or  because of the points to its left in .
Neither can the points in  dance, because Q is below them. So m_1=0.
On the other hand, the Case 2 analysis of the main column is still valid, the k_2 points in  to the right of Q being able to dance through to .

The situation is analogous if the main row satisfies Case 2 and the main column satisfies Case 1.
Table <ref> gives the total number of griddings for each possibility in Case 3.





Thus, for each ℓ1, the probability P_3(ℓ) = Case 3 and ℓ griddings is given by

    P_3(ℓ) 
    0 ,    ℓ=1 , 
    
    p_2b(ℓ-22)   q_1 +  p_1  q_2b(ℓ-22)  = λ^ℓ/2  (  +  )(1-)  (1-)
                            ,    ℓ2, even , 
    
    p_2a(ℓ-12)   q_1 +  p_1  q_2a(ℓ-12)   = 2 λ^(ℓ-1)/2  (1-)  (1-)
                            ,    ℓ3, odd ,

where q_1 is the probability that the main column satisfies Case 1,
formed from p_1
by replacing  and  with
 and ,
respectively.

We can now combine the three cases.
The asymptotic probability that the underlying permutation of a gridded permutation has exactly ℓ griddings is

    P_ℓ P_1(ℓ) + P_2(ℓ) + P_3(ℓ) .

After considerable simplification, facilitated by using a computer algebra system, we have

    P_ℓ   (1-)  (1-) ,    ℓ=1 ,
    ℓ λ^(ℓ-2)/2  (λ +  )   (  +  )2  (1-)  (1-) ,    ℓ2, even , 
    ℓ λ^(ℓ-3)/2  (λ +  )   (  + λ  )2  (1-)  (1-) ,    ℓ3, odd .


Finally, after further simplification, this gives us the correction factor for the L_3 corner type:

    κ(L_3) ∑_ℓ1P_ℓ/ℓλ (1-λ)/(1-)  (1-)λ (1-λ)/(1-(c-1)λ)  (1-(r-1)λ) .


8


  §.§.§ Corner type T4




.125
-2232103-1,1,-10,-1
3.67pt-2232101530,1-1,1,-10,-1 


This corner type has one peak and one tee.
It can't occur in an -shaped class.

Dancing at the peak depends on the points adjacent to the column divider
to the right of the corner,
whereas
tee dancing depends on the points adjacent to the column divider
to the left of the corner and points adjacent to the row divider below the corner.
So these
are asymptotically independent.
Thus, from the analysis for L_1 and L_3, and by Observation <ref>, we have

    κ(T_4)
    κ(L_1)  κ(L_3)
    1/2(1+r αλ/α + β) λ (1-λ)/(1-(c-1)λ)  (1-(r-1)λ) .


7


  §.§.§ Corner type X4




.125-2232101530,-1-1,1,-10,-1 

This corner type has two tees.
It can't occur in an -shaped or -shaped class.

Dancing at one of the tees depends on the points adjacent to the column divider
to the right of the corner and the row divider above the corner,
whereas
dancing at the other tee depends on the points adjacent to the column divider
to the left of the corner and points adjacent to the row divider below the corner.
So these
are asymptotically independent.
Thus, from the analysis for L_3, and by Observation <ref>, we have

    κ(X_4)
    κ(L_3)^2
    ( λ (1-λ)/(1-(c-1)λ)  (1-(r-1)λ))^2.






 §.§ Comparison of corner types


The correction factors induce a partial order on the corner types, in which
corner type τ_1 dominates corner type τ_2
if κ(τ_1)>κ(τ_2) for all values of r and c.
If this is the case, then
a class with smaller asymptotic size is the result whenever
a corner of type τ_1 is replaced with one of type τ_2 without changing the dimensions or the number of non-corner peaks.

The Hasse diagram of the corner type poset is shown in
Figure <ref>.
For a dashed edge,
strict ordering is only satisfied  under the condition specified, with the correction factors being equal otherwise.
For example, if r=1, then κ(T_2)=κ(L_4).







 §.§ Constrained gridded permutations


Recall that, if (M) is a connected one-corner class, then an M-gridded permutation σ^# is M-constrained if

  
  (a) every M-gridding of its underlying permutation σ is the result of zero or more points of σ^# dancing at a peak or diagonally or through a tee,
  and
  
  (b) in every M-gridding of σ, each non-blank cell contains at least two points.


Suppose a gridded permutation satisfies Part (a) of this definition.
Then, in order to satisfy Part (b),
it is sufficient that, in each non-blank cell, there are at least two points that can't dance,
since these points will be in the same cell in all griddings of the underlying permutation.

Non-corner peak points can always dance.
However, the set of points which can dance at the corner is determined by the position of the controllers.

For peak dancing (L_1) and diagonal dancing (L_7), each controller is an extremal (lowest, highest, leftmost or rightmost) point in one of the cells.
For diagonal dancing, the points which can dance are those that are in one of the cells adjacent to the corner and lie between the relevant controller and cell divider.

For tee dancing (L_3), each controller is either an extremal point in a cell, or else is the second lowest or second highest point in the corner cell.
As with diagonal dancing, points between a controller and the corresponding cell divider can dance.
However, there may also be a single point that is not between the controller and cell divider that can dance.
In addition, one or both of the extremal points in the corner cell may also be able to dance.



See Figure <ref> for an example of a gridded permutation in a class with L_3 corner type and a non-corner peak:
in each non-blank cell there are at least two points that can't dance.
Note that we may need four points in a cell above a controller in the main row to guarantee two points that can't dance,
since one point just above the controller may be able to dance, and the highest point may be a peak point that can dance.

In this context, we make the following definition.
Given a gridded permutation,
if C_1 and C_2 are two distinct cells in the same row, then they are interlocked if

  
  * C_1 contains at least four points above the second lowest point in C_2,
  
  * C_2 contains at least four points above the second lowest point in C_1,
  
  * C_1 contains at least four points below the second highest point in C_2, and
  
  * C_2 contains at least four points below the second highest point in C_1.

Similarly, if C_1 and C_2 are two distinct cells in the same column, then they are interlocked if

  
  * C_1 contains at least four points to the right of the second leftmost point in C_2,
  
  * C_2 contains at least four points to the right of the second leftmost point in C_1,
  
  * C_1 contains at least four points to the left of the second rightmost point in C_2, and
  
  * C_2 contains at least four points to the left of the second rightmost point in C_1.

Note that if two adjacent cells are interlocked, then their contents together don't form an increasing or decreasing sequence.

The following proposition gives sufficient conditions for a gridded permutation in a connected one-corner class to be constrained.


  Suppose (M) is a connected one-corner class and σ^#∈(M) is such that
  each pair of cells in the main row is interlocked and
  each pair of cells in the main column is interlocked.
  Then σ^# is M-constrained.


Suppose we remove the row dividers from σ^#, merging the cells in the main row into a single “super cell”.
This forms a horizontal skinny gridded permutation in which the contents of one cell are not monotone.
Let us denote this σ^.
Similarly, if we remove the column dividers, the result is a vertical skinny gridded permutation σ^= with one non-monotone cell.
We now use an argument similar to that employed in the proof of Proposition <ref> for skinny classes.

In any M-gridding of σ, by the monotonicity constraints, there must be
a column divider between each pair of adjacent monotone cells of σ^ that have the same orientation,
and also a column divider adjacent to each peak point of σ^ (where peaks are formed from two monotone cells).
Similarly, there must be
a row divider between each pair of adjacent monotone cells of σ^= that have the same orientation,
and also a row divider adjacent to each peak point of σ^=.
Thus, in any M-gridding of σ, there must be a divider between each pair of adjacent non-corner cells of σ^# that have the same orientation
and a divider adjacent to each non-corner peak point.

What of the dividers adjacent to the corner cell?
For this, we require a case analysis of each corner type.
But this is exactly what is presented in Sections <ref> to <ref>.
The interlocking of cells in σ^# guarantees the existence of the controllers, which restrict the possible griddings to those that result from zero or more points of σ^# dancing at a peak or diagonally or through a tee.

Moreover, in each non-blank cell there are at least two points that can't dance.
So each cell of any M-gridding of σ contains at least two points.


Finally, we prove that almost all gridded permutations in a connected one-corner class are constrained.


If (M) is a connected one-corner class, then almost all M-gridded permutations are M-constrained:

    σ^# is M-constrained : σ^#∈_n(M) 1 .



It is sufficient to prove that almost all M-gridded permutations satisfy the conditions of Proposition <ref>.
By Theorem <ref>, we know that the number of points in each non-blank cell of most n-point M-gridded permutations grows with n.
So, given any pair of cells both in the main row or both in the main column, the probability that they are interlinked in an n-point M-gridded permutation converges to 1 as n tends to infinity.





 §.§ Beyond connected one-corner classes


We conclude with some brief notes on extending our approach beyond ,  and -shaped classes.
Firstly, for any connected class, the asymptotic distribution of points between the cells in a typical gridded permutation can be established using the method presented in Section <ref>.
Secondly, for any connected class whose cell graph is either acyclic or unicyclic,
the approach of Section <ref> using generating functions can be extended to give
the asymptotic number of gridded permutations. For further details, see <cit.>.



Connected two-corner classes with no adjacent corner cells, such as the three shown in Figure <ref>, don't support any new possibilities for dancing, and are thus directly amenable to the analysis presented above.
If corner orientations are restricted to northeast and southwest, then we have what we call a snake, such as the class at the right of Figure <ref>.
The asymptotic enumeration of snakes with no adjacent corners also requires no additional analysis.



However, in general, connected two-corner classes with corners in adjacent cells, such as the semi-skinny (two row) classes shown in Figure <ref>, may exhibit non-corner diagonals.

Dancing at non-corner diagonals interacts with the dancing at the corners, so additional analysis would be required.
With three or more corners, the general situation quickly gets more complicated.
With four or more corners, the cell graph may be unicyclic.
See <cit.> for a detailed investigation of the structural complexity of unicyclic classes.

In general, disconnected classes are not directly amenable to our approach, because typical large gridded permutations in such classes may not be constrained.
For example, in 30,0,-11,1, the asymptotic expected number of points in the non-blank cell at the upper right is finite.
Indeed, the asymptotic probability of that cell being empty or containing just a single point is positive.
Further analysis would be required to handle this sort of situation.


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