# A pipe dream perspective on totally symmetric self-complementary plane
partitions

Daoji Huang and Jessica Striker

###### Abstract.

We characterize totally symmetric self-complementary plane partitions (TSSCPP)
as bounded compatible sequences satisfying a Yamanouchi-like condition. As
such, they are in bijection with certain pipe dreams. Using this
characterization and the recent bijection of [Gao-Huang] between reduced pipe
dreams and reduced bumpless pipe dreams, we give a bijection between
alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We
also give a different bijection in the 1432- and 2143-avoiding case that
preserves natural poset structures on the associated pipe dreams and bumpless
pipe dreams.

##  1\. Introduction

_Plane partitions_ are three-dimensional analogues of ordinary partitions.
Just as partitions in an $a\times b$ are counted by a lovely formula
$\binom{a+b}{a}$, plane partitions in an $a\times b\times c$ box are
enumerated by MacMahon’s product formula
$\displaystyle\prod_{i=1}^{a}\displaystyle\prod_{j=1}^{b}\displaystyle\prod_{k=1}^{c}\displaystyle\frac{i+j+k-1}{i+j+k-2}$
[Mac04]. In a 1986 [Sta86], Stanley considered symmetry operations on plane
partitions, namely, reflection (transpose), rotation, and complementation.
This yielded 10 symmetry classes of plane partitions consisting of plane
partitions invariant under combinations of these operations. The plane
partitions invariant under all three operations are called _totally symmetric
self-complementary_ (_TSSCPP_). As in the case of all plane partitions, each
symmetry class has a nice enumeration. The set of TSSCPP inside a $2n\times
2n\times 2n$ box was shown in 1994 by Andrews [And94] to be counted by
$\displaystyle\prod_{j=0}^{n-1}\displaystyle\frac{(3j+1)!}{(n+j)!}$ . This
was, at the time, the conjectured [MRR83] number of $n\times n$ _alternating
sign matrices_ (_ASM_). The 1996 proofs of this conjecture [Zei96, Kup96]
sparked a search for a natural, explicit bijection between TSSCPP and ASM.
Partial bijections have been found on small subsets, including the permutation
case [Str18], the case of two monotone triangle diagonals [CB12, Bet19], and
the $312$-avoiding case [ACGB11]. This paper interprets TSSCPP as _pipe
dreams_ to extend the bijection of [Str18] to what appears to be a larger
subset than any previous partial bijection; see Section 6 for discussion.

Our first main theorem is below; see Figure 1 for an example and Section 2 for
the relevant definitions. Given $\pi\in S_{n}$, let
$\operatorname{TSSCPP}^{red}(\pi)$ denote the set of TSSCPP whose associated
pipe dream is _reduced_ and has permutation $\pi$, and let
$\operatorname{ASM}^{red}(\pi)$ denote the set of ASM whose associated
_bumpless pipe dream_ is reduced and has permutation $\pi$.

######  Theorem 1.1.

Let $\pi\in S_{n}$. There is an explicit weight-preserving injection $\varphi$
from $\operatorname{TSSCPP}^{red}(\pi)$ to $\operatorname{ASM}^{red}(\pi)$. If
$\pi$ avoids $1432$, then $\varphi$ is a bijection.

While the bijection of Theorem 1.1 preserves a meaningful weight on both
sides, it does not, in general, preserve the natural partial order. A
corollary of our second main result, Theorem 5.20, gives a different poset-
preserving bijection between $\operatorname{TSSCPP}^{red}(\pi)$ and
$\operatorname{ASM}(\pi)$ in the case that $\pi$ avoids both $1432$ and
$2143$. (Note in this case,
$\operatorname{ASM}^{red}(\pi)=\operatorname{ASM}(\pi)$.) Theorem 5.20 itself
relates the posets $\operatorname{\mathrm{Slide}}(\pi)$ on pipe dreams and
$\operatorname{\mathrm{Droop}}(\pi)$ on bumpless pipe dreams of such
permutations, giving a poset-preserving bijection by decomposing into
Grassmannian and inverse-Grassmannian blocks.

The paper is organized as follows. Section 2 contains background on the
relevant objects, including the permutation case TSSCPP bijection of [Str18]
and the bijection of [GH21] between reduced pipe dreams and reduced bumpless
pipe dreams, which are important ingredients in our proof of Theorem 1.1.
Section 3 proves Theorem 3.3 characterizing TSSCPP as pipe dreams subject to a
Yamanouchi-like condition. Section 4 concerns Theorem 1.1 and its proof.
Section 5 proves Theorems 5.9, 5.12, and 5.20 relating the posets
$\operatorname{\mathrm{Droop}}(\pi)$ and $\operatorname{\mathrm{Slide}}(\pi)$
in the respective cases where $\pi$ is inverse-Grassmannian, Grassmannian, or
avoiding both $1432$ and $2143$. These theorems yield Corollaries 5.10, 5.14,
and 5.21, which give poset-preserving bijections between
$\operatorname{TSSCPP}^{red}(\pi)$ and $\operatorname{ASM}(\pi)$ for these
three types of permutations. Section 6 gives some concluding remarks.

$\Leftrightarrow$ $\Leftrightarrow$ $\Leftrightarrow$
$\left(\begin{array}[]{rrrrrr}1&0&0&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&1&-1&1&0\\\
0&0&0&1&0&0\\\ 0&1&0&-1&0&1\\\ 0&0&0&1&0&0\end{array}\right)$

Figure 1. An example of the bijection of this paper. From left to right the
objects are: TSSCPP, pipe dream, bumpless pipe dream, ASM. The pipe dream and
bumpless pipe dream both have permutation $135264$, which avoids $1432$. Note
the black rhombi in column $k$ (from the left) of the TSSCPP fundamental
domain correspond to the cross tiles in row $k$ (from the top) of the pipe
dream. This equals the number of blank tiles in row $k$ of the bumpless pipe
dream, which is the number of positive inversions of row $k$ of the ASM.

##  2\. Background

In this section, we review relevant definitions and bijections from the
literature. Subsections 2.1, 2.2, 2.4, and 2.5 review definitions of ASM,
TSSCPP, bumpless pipe dreams, and pipe dreams, respectively. Subsections 2.3
and 2.6 contain less-familiar bijections that are important for our main
results.

###  2.1. Alternating sign matrices

In this subsection, we define alternating sign matrices (see e.g. [MRR83]) and
the weight that is preserved in Theorem 1.1.

######  Definition 2.1.

An alternating sign matrix (ASM) is a square matrix with entries in
$\\{0,1,-1\\}$ such that the rows and columns each sum to $1$ and the nonzero
entries alternate in sign across each row and across each column.

Alternating sign matrices are in bijection with configurations of the _six-
vertex / square ice model_ of statistical physics with _domain wall boundary
conditions_ ; this was an essential element of the enumeration proof of
[Kup96]. The $3\times 3$ alternating sign matrices are below.

$\footnotesize\left(\begin{array}[]{rrr}1&0&0\\\ 0&1&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{rrr}1&0&0\\\ 0&0&1\\\
0&1&0\end{array}\right)\left(\begin{array}[]{rrr}0&1&0\\\ 0&0&0\\\
0&0&1\end{array}\right)\left(\begin{array}[]{rrr}0&1&0\\\ 1&-1&1\\\
0&1&0\end{array}\right)\left(\begin{array}[]{rrr}0&1&0\\\ 0&0&1\\\
1&0&0\end{array}\right)\left(\begin{array}[]{rrr}0&0&1\\\ 1&0&0\\\
0&1&0\end{array}\right)\left(\begin{array}[]{rrr}0&0&1\\\ 0&1&0\\\
1&0&0\end{array}\right)$

In Figure 2, left and center-left are an alternating sign matrix and its
corresponding square ice configuration; the horizontal molecules correspond to
$+1$, the vertical molecules correspond to $-1$, and all other molecules
correspond to $0$. Center-right is its six-vertex configuration, where the six
molecule configurations are replaced by directed edges. Figure 2, right, shows
the corresponding _bumpless pipe dream_ , which will be discussed shortly.

$\left(\begin{array}[]{rrrr}0&1&0&0\\\ 1&-1&0&1\\\ 0&0&1&0\\\
0&1&0&0\end{array}\right)$

Figure 2. An alternating sign matrix and its corresponding square ice
configuration, six-vertex configuration, and bumpless pipe dream.

An important statistic on an alternating sign matrix $A$ is the positive
inversion number:

$\nu(A)=\sum_{1\leq i<k<n}\sum_{1\leq\ell\leq k\leq n}A_{ij}A_{k\ell}.$

The positive inversion number of $A$ equals the number of  vertices in the
corresponding six-vertex configuration. Equivalently, $\nu(A)$ equals the
number of entries $A_{ij}=0$ such that
$\sum_{i^{\prime}=1}^{i}A_{i^{\prime}j}=\sum_{j^{\prime}=1}^{j}A_{ij^{\prime}}=0$.
Let $\operatorname{NW}(A)$ denote the set of matrix indices $(i,j)$ of all
such entries. We use the notation $\operatorname{NW}(A)$ since these $0$
entries are precisely those whose first nonzero entry to their right is a $1$
and the first nonzero entry below is a $1$; that is, they are north and west
of $1$ entries. See for instance [Beh13, Sections 2.1 and 5.2] for these well-
known correspondences. The weight that is preserved in our main bijection is
the following refinement of the positive inversion number statistic by row:
$\operatorname{wt}(A):=\prod_{(i,j)\in\operatorname{NW}(A)}x_{i}$. For the ASM
in Figure 2, $\operatorname{wt}(A)=x_{1}x_{2}x_{3}$.

###  2.2. Totally symmetric self-complementary plane partitions

In this subsection, we define plane partitions and their symmetry classes (see
e.g. [Sta86]). The weight that is preserved in Theorem 1.1 will be discussed
in Subsection 2.3.

######  Definition 2.2.

A plane partition $t$ is a rectangular array of nonnegative integers
$(t_{i,j})_{i,j\geq 1}$ such $t_{i,j}\geq t_{i^{\prime},j^{\prime}}$ if $i\leq
i^{\prime}$, $j\leq j^{\prime}$. We say $t$ is in an $a\times b\times c$
bounding box if $t_{i,j}=0$ whenever $i>a$ or $j>b$ and $t_{i,j}\leq c$ for
all $i,j$. Let $PP(a\times b\times c)$ denote the set of plane partitions in
an $a\times b\times c$ bounding box.

######  Remark 2.3.

We can also view $t\in PP(a\times b\times c)$ as a finite set of positive
integer lattice points $(i,j,k)$ with $1\leq i\leq a$, $1\leq j\leq b$, and
$1\leq k\leq c$ such that if $(i,j,k)\in t$ and $1\leq i^{\prime}\leq i$,
$1\leq j^{\prime}\leq j$, $1\leq k^{\prime}\leq k$ then
$(i^{\prime},j^{\prime},k^{\prime})\in t$. This well-known bijection is given
as $(i,j,k)\in t$ if and only if $t_{i,j}\geq k$. We will use both of these
characterizations in the next definition.

######  Definition 2.4.

A plane partition $t$ is symmetric if $t_{i,j}=t_{j,i}$ for all $i,j$. $t$ is
cyclically symmetric if whenever $(i,j,k)\in t$ then $(j,k,i)\in t$ as well.
$t$ is totally symmetric if it is both symmetric and cyclically symmetric, so
that whenever $(i,j,k)\in t$ then all six permutations of $(i,j,k)$ are also
in $t$. The complement $t^{C}$ of $t$ inside a given bounding box $a\times
b\times c$ is defined as $t^{C}_{i,j}=c-t_{a-i+1,b-j+1}$ for all $1\leq i\leq
a,1\leq j\leq b$. That is, $t^{C}_{i,j}$ equals the number of empty cubes
above $t_{a-i+1,b-j+1}$ in the bounding box. A plane partition $t$ is self-
complementary inside a given bounding box if $t=t^{C}$. A totally symmetric
self-complementary plane partition (TSSCPP) is a plane partition which is both
totally symmetric and self-complementary.

Note that for there to exist a self-complementary plane partition in an
$a\times b\times c$ bounding box, the volume $abc$ of the box must be an even
number. In addition, cyclic symmetry requires $a=b=c$, therefore, we need
$a=b=c=2n$ for there to exist a TSSCPP inside an $a\times b\times c$ bounding
box.

######  Definition 2.5.

Let $\operatorname{TSSCPP}(n)$ denote the set of TSSCPP inside a $2n\times
2n\times 2n$ box.

###  2.3. TSSCPP boolean triangles and a permutation case bijection

In this subsection, we review the characterization from [Str18] of TSSCPP as
boolean triangles and the bijection of the same paper between permutation
matrices and TSSCPP boolean triangles whose entries weakly decrease along
rows. We also describe the weight on TSSCPP preserved in Theorem 1.1.

######  Definition 2.6 (Def 2.12 of [Str18]).

A TSSCPP boolean triangle of order $n$ is a triangular integer array
$b=\\{b_{i,j}\\}$ for $1\leq i\leq n-1$, $n-i\leq j\leq n-1$ with entries in
$\\{0,1\\}$ such that the diagonal partial sums satisfy the following
inequality for all $1\leq j<i\leq n-1$:

(1) $1+\sum_{k=j+1}^{i}b_{k,n-j-1}\geq\sum_{k=j}^{i}b_{k,n-j}.$

Call this the $(i,j)$-inequality, in which $n-j$ and $n-j-1$ are the diagonals
being compared and $i$ indicates the row index of where the sums stop.

We give below the indexing of a generic TSSCPP boolean triangle.

$\begin{array}[]{ccccccccccc}&&&&&b_{1,n-1}&&&&&\\\
&&&&b_{2,n-2}&&b_{2,n-1}&&&&\\\ &&&b_{3,n-3}&&b_{3,n-2}&&b_{3,n-1}&&&\\\
&&&&&\vdots&&&&&\\\
&&b_{n-1,1}&&b_{n-1,2}&\cdots&b_{n-1,n-2}&&b_{n-1,n-1}&&\end{array}$

Below are a non-example and an example of a TSSCPP boolean triangle.

$\begin{array}[]{ccccccccccc}&&&&&1&&&&&\\\ &&&&1&&1&&&&\\\ &&&1&&0&&0&&&\\\
&&1&&0&&0&&1&&\\\ &0&&0&&0&&1&&0&\end{array}$
$\begin{array}[]{ccccccccccc}&&&&&1&&&&&\\\ &&&&1&&1&&&&\\\ &&&1&&0&&0&&&\\\
&&1&&0&&1&&1&&\\\ &0&&0&&0&&0&&0&\end{array}$

In the left triangle, the $(4,1)$-inequality is not satisfied, since
$\sum_{k=1}^{4}b_{k,n-1}=3$ while $\sum_{k=2}^{4}b_{k,n-2}=1$. In the triangle
on the right, all $(i,j)$-inequalities are satisfied.

######  Proposition 2.7 (Prop 2.13 of [Str18]).

TSSCPP boolean triangles of order $n$ are in bijection with
$\operatorname{TSSCPP}(n)$.

The bijection proceeds by taking the fundamental domain of the TSSCPP,
transforming it into a nest of non-intersecting lattice paths, and then
recording the two different types of steps in each path as $0$ and $1$. The
diagonal partial sum condition (1) is equivalent to the requirement that the
paths do not intersect. See [Str18] for details.

We now review the characterization of a certain subset of TSSCPP boolean
triangles.

######  Definition 2.8 (Def 3.1 of [Str18]).

A permutation TSSCPP boolean triangle is a TSSCPP boolean triangle with weakly
decreasing rows.

That is, the entries equal to one in a permutation TSSCPP boolean triangle are
all left-justified. The terminology ‘permutation’ in the above definition is
justified by the weight-preserving bijection in the theorem below. An example
of this bijection is given in Figure 3.

######  Theorem 2.9 (Theorem 3.5 of [Str18]).

There is a natural, statistic-preserving bijection between $n\times n$
permutation matrices with inversion number $p$ and permutation TSSCPP boolean
triangles of order $n$ with $p$ zeros.

$\begin{array}[]{c}\mbox{TSSCPP}\\\
\includegraphics[scale={.3}]{TFundDomain.pdf}\end{array}$ $\leftrightarrow$
$\begin{array}[]{c}\mbox{Permutation }\\\ \mbox{TSSCPP}\\\ \mbox{Boolean
triangle}\\\ \begin{array}[]{ccccccccccc}&&&&&1&&&&&\\\ &&&&0&&0&&&&\\\
&&&1&&1&&0&&&\\\ &&0&&0&&0&&0&&\\\ &1&&0&&0&&0&&0&\end{array}\end{array}$
$\leftrightarrow$ $\begin{array}[]{c}\mbox{Permutation}\\\ \mbox{Matrix}\\\
\mbox{ }\\\ \scalebox{0.9}{$\left(\begin{array}[]{cccccc}0&0&0&1&0&0\\\
0&0&0&0&0&1\\\ 0&0&1&0&0&0\\\ 0&0&0&0&1&0\\\ 1&0&0&0&0&0\\\
0&1&0&0&0&0\end{array}\right)$}\end{array}$

Figure 3. An example of the permutation case bijection of [Str18, Theorem 3.5]

The injection $\varphi$ in Theorem 1.1 extends this bijection under the mild
transformation of flipping the resulting matrix vertically (or reversing the
one-line notation of the permutation). Thus, in Theorem 1.1 we instead map the
TSSCPP in Figure 3, left, to the vertical reflection of the matrix in Figure
3, right.

The TSSCPP weight preserved in Theorem 1.1 is the number of $1$ entries in the
$k$th row from the bottom in its boolean triangle. More specifically, for
$T\in\operatorname{TSSCPP}(n)$ with boolean triangle $b$, let $\text{one}(T)$
denote the set of indices of the entries of $b$ that equal $1$. Then
$\operatorname{wt}(T):=\prod_{(i,j)\in\text{one}(T)}x_{n-i}$. For the TSSCPP
in Figure 3, $\operatorname{wt}(T)=x_{1}x_{3}^{2}x_{5}$ and for the TSSCPP in
Figure 5, $\operatorname{wt}(A)=x_{2}^{2}x_{3}x_{4}$. Note the weight can also
be seen directly on the TSSCPP fundamental domain, as each $1$ in row $n-k$ of
$b$ corresponds to a black rhombus in column $k$ (from the left) of the
fundamental domain, as shown in these figures.

###  2.4. Bumpless pipe dreams

In this subsection, we define bumpless pipe dreams and describe the bijection
with alternating sign matrices.

######  Definition 2.10.

A bumpless pipe dream [LLS21] of size $n$ is a tiling of an $n\times n$ grid
of squares by the following six types of tiles: , , , , , , such that $n$
pipes traveling from the south border to the east border are formed. We denote
the set of bumpless pipe dreams of size $n$ as $\operatorname{BPD}(n)$. We say
a bumpless pipe dream is reduced if no two pipes cross twice. We associate a
permutation to each reduced bumpless pipe dream by labeling the pipes
$1,\cdots,n$ from left to right on the south border and read off the pipe
labels from top to bottom on the east border. Let
$\operatorname{BPD}^{red}(\pi)$ denote the set of all reduced bumpless pipe
dreams with permutation $\pi$.

A simple droop is a move on a bumpless pipe dream bounded by a $2\times 2$
square, as shown below. The four pairs of $2\times 2$ squares show all four
possibilities a pipe enters and leaves the $2\times 2$ square.

For each reduced bumpless pipe dream $D$, we define its weight to be the
monomial $\operatorname{wt}(D):=\prod_{(i,j)\in\operatorname{blank}(D)}x_{i}$
where $\operatorname{blank}(D)$ denotes the set of the -tiles in $D$.

There is a natural, weight-preserving bijection between
$\operatorname{BPD}(n)$ and $\operatorname{ASM}(n)$, as described in [Wei21].
To obtain an ASM from a BPD we simply replace each  with a 1, each  with a
$-1$, and the other four allowed squares with 0s. The blank tiles  of a BPD
$D$ correspond to the NW zeros of the associated ASM $A$. Thus
$\operatorname{blank}(D)=NW(A)$, so
$\operatorname{wt}(D)=\operatorname{wt}(A)$. For the inverse map, it is not
difficult to see the positions of  and  uniquely determine a bumpless pipe
dream. See Figure 2 for an example.

###  2.5. Pipe dreams

In this subsection, we define pipe dreams and bounded compatible sequences and
describe the bijection between them.

######  Definition 2.11.

A pipe dream [BB93] of size $n$ is a tiling of an $n\times n$ grid of squares
with two kinds of tiles, the cross-tile  and elbow-tile , such that the
positions on or below the main (anti)diagonal only consist of elbow-tiles. We
think of a pipe dream as $n$ pipes, labelled $1,\cdots,n$ traveling from the
north border and exiting from the west border. We denote the set of pipe
dreams of size $n$ as $\operatorname{PD}(n)$. We say a pipe dream is reduced
if no two pipes cross twice. We associate a permutation to each reduced pipe
dream by reading from top to bottom the labels of each pipe on the west border
of the pipe dream. (One can also assign permutations to non-reduced pipe
dreams, but this will not be important for the present paper.) Let
$\operatorname{PD}^{red}(\pi)$ denote the set of reduced pipe dreams with
permutation $\pi$.

The set of pipe dreams for a fixed permutations are connected by _chute_ and
_ladder_ moves. For precise definitions see [BB93]. When a ladder (or chute)
move is bounded by a $2\times 2$ square, we call this move a simple slide, as
shown below.

Figure 4 shows the set of pipe dreams $\operatorname{PD}^{red}(1432)$. The
first four pipe dreams are connected by simple slides; the fifth is not.

Figure 4. $\operatorname{PD}^{red}(1432)$

For each pipe dream $D$, we define its weight to be the monomial
$\operatorname{wt}(D):=\prod_{(i,j)\in\operatorname{cross}(D)}x_{i}$ where
$\operatorname{cross}(D)$ denotes the set of the -tiles in $D$.

######  Definition 2.12.

A bounded compatible sequence [BJS93] is a pair $(\mathbf{a},\mathbf{r})$
where $\mathbf{a}=(a_{1},\cdots,a_{\ell})$ and
$\mathbf{r}=(r_{1},\cdots,r_{\ell})$ are words of positive integers,
satisfying the following conditions:

  1. (a)

$r_{1}\geq r_{2}\geq\cdots\geq r_{\ell}$,

  2. (b)

$a_{i}\geq r_{i}$ for all $1\leq i\leq\ell$,

  3. (c)

$r_{i}>r_{i+1}$ if $a_{i}\geq a_{i+1}$.

There is a simple bijection between $\operatorname{PD}(n)$ and the set of all
bounded compatible sequences where $a_{i}<n$ for each $i$, see [BB93]. Given a
bounded compatible sequence $(\mathbf{a},\mathbf{r})$, we may construct a pipe
dream by putting a cross-tile at position $(r_{i},a_{i}+1-r_{i})$ for each
$1\leq i\leq\ell$ and fill the remaining positions with elbow-tiles.
Conversely, given a pipe dream, we may construct a bounded compatible sequence
as follows: scan the pipe dream from bottom to top and within each row left to
right, and whenever we encounter a cross-tile at position $(r,c)$ we append
$(r+c-1,r)$ to the compatible sequence. For example, the corresponding bounded
compatible sequences for the pipe dreams in Figure 4 are as follows; the
vector $\mathbf{a}$ is recorded in the top row and $\mathbf{r}$ in the bottom
row.

$\left(\begin{smallmatrix}3&2&3\\\
3&2&2\end{smallmatrix}\right),\left(\begin{smallmatrix}3&2&3\\\
3&2&1\end{smallmatrix}\right),\left(\begin{smallmatrix}3&2&3\\\
3&1&1\end{smallmatrix}\right),\left(\begin{smallmatrix}3&2&3\\\
2&1&1\end{smallmatrix}\right),\left(\begin{smallmatrix}2&3&2\\\
2&2&1\end{smallmatrix}\right).$

###  2.6. Reduced BPD-PD bijection

Both reduced pipe dreams and reduced bumpless pipe dreams give combinatorial
formulas for Schubert polynomials $\mathfrak{S}_{\pi}$, $\pi\in S_{\infty}$
which are important polynomials in the study of Schubert calculus [BB93,
LLS21]. Explicitly,

$\mathfrak{S}_{\pi}=\sum_{D\in\operatorname{PD}^{red}(\pi)}\prod_{(r,c)\in\operatorname{cross}(D)}x_{r}=\sum_{D\in\operatorname{BPD}^{red}(\pi)}\prod_{(r,c)\in\operatorname{blank}(D)}x_{r}.$

For this reason, there exists a weight-preserving bijection between
$\operatorname{PD}^{red}(\pi)$ and $\operatorname{BPD}^{red}(\pi)$, where the
weight of a PD or BPD is its monomial contribution to the Schubert polynomial
indexed by its permutation.

In [GH21], such an explicit direct bijection
$\varphi:\operatorname{BPD}^{red}(\pi)\rightarrow\operatorname{PD}^{red}(\pi)$
is given using an iterative algorithm. To find the image of a BPD under
$\varphi$, the algorithm computes in each iteration the position of one
crossing in the corresponding PD. For a detailed description of this process,
see [GH21, Definition 3.1]. For explicit examples, see [GH21, Example 3.4].
This bijection is weight-preserving; in particular, for
$D\in\operatorname{BPD}^{red}(\pi)$, the number of blank tiles in row $k$
equals the number of cross-tiles in row $k$ of $\varphi(D)$.

Because the bijection is weight-preserving and there is a unique lowest weight
monomial that corresponds to the Lehmer code of the permutation in each
Schubert polynomial, the permutation BPD is mapped to the bottom pipe dream,
the unique pipe dream with all crosses left-justified.

##  3\. Characterizing TSSCPP as pseudo-Yamanouchi pipe dreams

This section focuses on our first theorem: a characterization of TSSCPP as a
subset of all (reduced and non-reduced) pipe dreams.

###  3.1. Mapping TSSCPP into pipe dreams

Recall the bijection of Proposition 2.7 from TSSCPP to the TSSCPP boolean
triangles of Definition 2.6. As TSSCPP boolean triangles are triangular arrays
with entries in $\\{0,1\\}$, we can transform them to pipe dreams (reduced and
non-reduced), since these are triangular arrays of tiles with two choices for
each spot. There are several possibilities for how to do this; we choose to
correspond each $1$ to a cross-tile  and each $0$ to an elbow-tile . There are
also several choices for orientation of the triangle. We set the following
convention.

Given a TSSCPP boolean triangle $b$ of order $n$, we create a triangular array
$y_{i,j}$, $1\leq i\leq n-1$, $1\leq j\leq n-i$ of zeros and ones where
$y_{i,j}=b_{n-i,i+j-1}$. That is, we flip $b$ vertically and justify to the
left.

$\begin{array}[]{ccccc}b_{n-1,1}&b_{n-1,2}&b_{n-1,3}&\cdots&b_{n-1,n-1}\\\
b_{n-2,2}&b_{n-2,3}&\cdots&b_{n-2,n-1}\\\ b_{n-3,3}&\cdots&b_{n-3,n-1}\\\
&\iddots\\\
b_{1,n-1}\end{array}\begin{array}[]{ccccc}y_{1,1}&y_{1,2}&y_{1,3}&\cdots&y_{1,n-1}\\\
y_{2,1}&y_{2,2}&\cdots&y_{2,n-2}&\\\ y_{3,1}&\cdots&y_{3,n-3}&&\\\
&\iddots&&&\\\ y_{n-1,1}\end{array}$

The inequality of Definition 2.6 translates to the following:

$1+\sum_{k=1}^{i}y_{j-k,k}\geq\sum_{k=1}^{i+1}y_{j-k+1,k}\text{ \ \ for all
}1\leq i<j\leq n-1.$

Now we turn each $1$ into a cross-tile  and each $0$ into an elbow-tile . We
call the pipe dreams that lie in this image the TSSCPP pipe dreams. Note that
permutation TSSCPP boolean triangles have weakly decreasing rows; this
corresponds to left-justified crosses in the associated pipe dream.

$\leftrightarrow$ $\scalebox{0.8}{$\begin{array}[]{ccccccccccc}&&&&&0&&&&&\\\
&&&&0&&1&&&&\\\ &&&1&&0&&0&&&\\\ &&1&&0&&1&&0&&\\\
&0&&0&&0&&0&&0&\end{array}$}\leftrightarrow\hskip
5.0pt\begin{matrix}0&0&0&0&0\\\ 1&0&1&0&\\\ 1&0&0&&\\\ 0&1&&&\\\
0&&&&\end{matrix}\leftrightarrow$

Figure 5. An example of transforming a TSSCPP to a pipe dream. Note the weight
of this TSSCPP is $x_{2}^{2}x_{3}x_{4}$.

###  3.2. A Yamanouchi-like condition on bounded compatible sequences

In this subsection, we prove Theorem 3.3 characterizing TSSCPP pipe dreams. We
also prove Lemma 3.2, which will be used in Section 4.

######  Definition 3.1.

Given a bounded compatible sequence

$(\mathbf{a},\mathbf{r})=((a_{1},\cdots,a_{\ell}),(r_{1},\cdots,r_{\ell})),$

define $\operatorname{count}(k,j)(\mathbf{a})$ (or $\operatorname{count}(k,j)$
when $\mathbf{a}$ is understood) to be the number of $j$ that appear in
$a_{1},\cdots,a_{k}$. We say that $(\mathbf{a},\mathbf{r})$ is pseudo-
Yamanouchi if for all $1\leq k\leq\ell$, $1\leq j\leq n-2$,
$1+\operatorname{count}(k,j)\geq\operatorname{count}(k,j+1)$. We also say that
a pipe dream is pseudo-Yamanouchi if its corresponding bounded compatible
sequence is so.

######  Lemma 3.2.

For any $\pi$, the bottom pipe dream is pseudo-Yamanouchi.

###### Proof.

The bottom pipe dream is the unique pipe dream of $\pi$ with all left-
justified cross-tiles. Thus the bounded compatible sequence
$(\mathbf{a},\mathbf{r})$ is either empty or $\mathbf{a}$ is made up of
increasing runs such that it can be written for some $m\geq 1$ as

$\mathbf{a}=(j_{1},j_{1}+1,\ldots,j_{1}^{*}-1,j_{1}^{*},j_{2},j_{2}+1,\ldots,j_{2}^{*}-1,j_{2}^{*},\ldots,j_{m},j_{m}+1,\ldots,j_{m}^{*}-1,j_{m}^{*})$

where $j_{1}>j_{2}>\cdots>j_{m}$ and $j_{i}^{*}\geq j_{i}$ for all $1\leq
i\leq m$. Because $j_{1}>j_{2}>\cdots>j_{m}$, each increasing run needs to
start with a smaller number than the previous.

Suppose $(\mathbf{a},\mathbf{r})$ is not pseudo-Yamanouchi. Choose the
smallest $k$ such that there exists a $j$ for which
$1+\operatorname{count}(k,j)<\operatorname{count}(k,j+1)$. Since
$\operatorname{count}(k,j)$ is a non-negative increasing function of $k$, it
must be that $1+\operatorname{count}(k-1,j)=\operatorname{count}(k-1,j+1)$ and
$a_{k}=j+1$, since we chose $k$ to be the smallest value with the property. If
we are in row $j+1$ ($r_{k}=j+1$), then this is the first time that $j+1$ has
appeared in $\mathbf{a}$, so $\operatorname{count}(k,j+1)=1$ and thus cannot
be greater than $1+\operatorname{count}(k,j)$. If $r_{k}>j+1$, then
$a_{k-1}=j$, since the cross-tiles are left-justified, and thus
$\operatorname{count}(k-1,j)+1=\operatorname{count}(k,j)$. But
$1+\operatorname{count}(k-1,j)\geq\operatorname{count}(k-1,j+1)=\operatorname{count}(k,j+1)-1$.
So finally, $\operatorname{count}(k,j)\geq\operatorname{count}(k,j+1)-1$,
which is a contradiction. ∎

######  Theorem 3.3.

$\operatorname{TSSCPP}(n)$ is in weight-preserving bijection with the set of
pseudo-Yamanouchi pipe dreams in $\operatorname{PD}(n)$.

###### Proof.

We identify a TSSCPP with the 0-1 triangular array $(y_{i,j})_{1\leq i\leq
n-1,1\leq j\leq n-i}$ satisfying the inequalities
$1+\sum_{k=1}^{i}y_{j-k,k}\geq\sum_{k=1}^{i+1}y_{j-k+1,k}$ for all $1\leq
i<j\leq n-1$, as described in Section 3.1. These inequalities mean the
following: for any position $(i,j)$ in the corresponding pipe dream, the
number of crosses in the same diagonal as $(i,j)$ at or below row $i$ can be
at most one more than the number of crosses in the previous diagonal at or
below row $i$. Therefore it suffices to check this property when $(i,j)$ is a
cross to decide whether the pipe dream is TSSCPP.

Now suppose $(a,r)$ is an entry of a pseudo-Yamanouchi compatible sequence.
Then by the definition of the reading order, all crosses that appear at or
below row $r$ in the $(a-1)$st diagonal of the corresponding pipe dream appear
before $(a,r)$ in the compatible sequence. Therefore the inequality for the
$(r,a-r+1)$ position is implied by the pseudo-Yamanouchi property. The
converse is true by a similar argument.

By definition, $\operatorname{wt}(T)=\operatorname{wt}(D)$, where $T$ is a
TSSCPP and $D$ its corresponding pseudo-Yamanouchi pipe dream. So this
bijection is weight-preserving. ∎

See Figure 5 for an example.

##  4\. A bijection between TSSCPP and ASM in the reduced, 1432-avoiding case

In this section, we prove our first main result, Theorem 1.1. The proof uses
the following theorem and lemmas, the first of which is due to Yibo Gao.

We need the following terminology. A permutation $\pi$ avoids a permutation
$\pi^{\prime}$ if there is no subsequence of $\pi$ having the same relative
order as $\pi^{\prime}$.

######  Theorem 4.1 ([Gao21, Theorem 4.1]).

If $\pi\in S_{n}$ avoids $1432$, then any two reduced pipe dreams of $\pi$ are
connected by simple slides.

######  Lemma 4.2.

Suppose $D\in\operatorname{PD}(n)$ is pseudo-Yamanouchi and
$D^{\prime}\in\operatorname{PD}(n)$ is related to $D$ by a simple slide. Then
$D^{\prime}$ is pseudo-Yamanouchi.

###### Proof.

Suppose $D\in\operatorname{PD}(n)$ is pseudo-Yamanouchi. Let
$(\mathbf{a},\mathbf{r})=((a_{1},\cdots,a_{\ell}),(r_{1},\cdots,r_{\ell}))$ be
its associated bounded compatible sequence. Suppose for some $1<i<n$, $D$ has
a  tile at position $(r_{i},a_{i}+1-r_{i})$ and no  tiles at positions
$(r_{i},a_{i}+2-r_{i})$, $(r_{i-1},a_{i}-r_{i})$, or
$(r_{i-1},a_{i}+1-r_{i})$. Then a simple slide may be applied to $D$,
resulting in another pipe dream $D^{\prime}$ with  tile at position
$(r_{i-1},a_{i}+1-r_{i})$ and no  tiles at positions $(r_{i},a_{i}+1-r_{i})$,
$(r_{i},a_{i}+2-r_{i})$, or $(r_{i-1},a_{i}-r_{i})$. That is, the simple slide
moves the  tile up one unit and to the right one unit and there were no other
tiles in these intermediate squares. This preserves the diagonal but
decrements the row index, creating a new bounded compatible sequence
$(\mathbf{a}^{\prime},\mathbf{r}^{\prime})=((a_{1}^{\prime},\cdots,a_{\ell}^{\prime}),(r_{1}^{\prime},\cdots,r_{\ell}^{\prime}))$
such that

(2) $a_{k}^{\prime}=\begin{cases}a_{k}&k<i\\\ a_{k+1}&i\leq k<m\\\
a_{i}&k=m\\\ a_{k}&m<k\leq\ell\end{cases}$

where $m>i$ is uniquely chosen so that
$(\mathbf{a}^{\prime},\mathbf{r}^{\prime})$ satisfies the conditions of a
bounded compatible sequence. Let $\tilde{j}:=a_{i}$. So $\mathbf{a}$ and
$\mathbf{a}^{\prime}$ differ only in that $\tilde{j}$ has slid to the right
from index $i$ to $m$.

Recall $\operatorname{count}(k,j)(\mathbf{a})$ denotes the number of $j$ that
appear in $a_{1},\cdots,a_{k}$. By Theorem 3.3, $(\mathbf{a},\mathbf{r})$ is
pseudo-Yamanouchi, so for all $1\leq k\leq\ell$, $1\leq j\leq n-2$,
$1+\operatorname{count}(k,j)(\mathbf{a})\geq\operatorname{count}(k,j+1)(\mathbf{a})$.
We need only check that $(\mathbf{a}^{\prime},\mathbf{r}^{\prime})$ is also
pseudo-Yamanouchi. The only values of $j$ we need to consider are
$\tilde{j}-1$, $\tilde{j}$, and $\tilde{j}+1$.

By (2),
$\operatorname{count}(k,j)(\mathbf{a}^{\prime})=\operatorname{count}(k,j)(\mathbf{a})$
for all values of $j$ when $k<i$ or $k\geq m$. Thus we need only check the
pseudo-Yamanouchi inequality for $k$ in the range $i\leq k<m$.

Suppose $i\leq k<m$. Since $(\mathbf{a},\mathbf{r})$ and
$(\mathbf{a}^{\prime},\mathbf{r}^{\prime})$ are related by a simple slide, we
know there is no cross in $D$ at position $(r_{i-1},a_{i}-r_{i})$. That is,
$a_{m-1}\neq\tilde{j}-1$. Furthermore, $a_{k}\neq\tilde{j}-1$ for all $i\leq
k<m$. Thus
$\operatorname{count}(k,\tilde{j}-1)(\mathbf{a}^{\prime})=\operatorname{count}(k,\tilde{j}-1)(\mathbf{a})$
for all $i\leq k<m$ while
$\operatorname{count}(k,\tilde{j})(\mathbf{a}^{\prime})=\operatorname{count}(k,\tilde{j})(\mathbf{a})-1$
in this same range. So

$\operatorname{count}(k,\tilde{j}-1)(\mathbf{a}^{\prime})=\operatorname{count}(k,\tilde{j}-1)(\mathbf{a})\geq\operatorname{count}(k,\tilde{j})(\mathbf{a})-1=\operatorname{count}(k,\tilde{j})(\mathbf{a}^{\prime}).$

So the pseudo-Yamanouchi condition is more than satisfied when comparing
diagonals $\tilde{j}-1$ and $\tilde{j}$.

Since $(\mathbf{a},\mathbf{r})$ and
$(\mathbf{a}^{\prime},\mathbf{r}^{\prime})$ are related by a simple slide, we
also know there is no cross in $D$ at position $(r_{i},a_{i}+2-r_{i})$. That
is, $a_{i+1}\neq\tilde{j}+1$. Furthermore, $a_{k}\neq\tilde{j}+1$ for all
$i\leq k<m$. Thus
$\operatorname{count}(k,\tilde{j}+1)(\mathbf{a}^{\prime})=\operatorname{count}(k,\tilde{j}+1)(\mathbf{a})$
for all $i\leq k<m$ while
$\operatorname{count}(k,\tilde{j})(\mathbf{a}^{\prime})=\operatorname{count}(k,\tilde{j})(\mathbf{a})-1$
in this same range. So

$1+\operatorname{count}(k,\tilde{j})(\mathbf{a}^{\prime})=1+\operatorname{count}(k,\tilde{j})(\mathbf{a})\geq\operatorname{count}(k,\tilde{j}+1)(\mathbf{a})=\operatorname{count}(k,\tilde{j}+1)(\mathbf{a}^{\prime}).$

Thus, the pseudo-Yamanouchi condition is satisfied on diagonals $\tilde{j}$
and $\tilde{j}+1$.

Therefore, $(\mathbf{a}^{\prime},\mathbf{r}^{\prime})$ is pseudo-Yamanouchi,
implying $D^{\prime}$ is pseudo-Yamanouchi. ∎

######  Lemma 4.3.

If $\pi\in S_{n}$ avoids $1432$, then all reduced pipe dreams of $\pi$ are
pseudo-Yamanouchi.

###### Proof.

Choose $\pi\in S_{n}$ that avoids $1432$. Using the previous two lemmas, we
know that simple slides preserve the pseudo-Yamanouchi property and that all
reduced pipe dreams in $\operatorname{PD}^{red}(\pi)$ are connected by simple
slides. So we need only show one reduced pipe dream is pseudo-Yamanouchi, and
then all of them are. By Lemma 3.2, the bottom (permutation) pipe dream is
pseudo-Yamanouchi. Thus the lemma is proved. ∎

###### Proof of Theorem 1.1.

Let $\pi\in S_{n}$. The explicit bijection
$\varphi:\operatorname{PD}^{red}(\pi)\rightarrow\operatorname{BPD}^{red}(\pi)$
of [GH21] discussed in Section 2.6 is weight-preserving; in particular, for
$D\in\operatorname{PD}^{red}(\pi)$, the number of cross-tiles in row $k$
equals the number of blank tiles in row $k$ of $\varphi(D)$, so
$\operatorname{wt}(D)=\operatorname{wt}(\varphi(D))$. By Theorem 3.3, TSSCPP
are characterized as the set of pseudo-Yamanouchi pipe dreams in
$\operatorname{PD}(n)$, and the weight is preserved in this bijection. Thus
whenever such pipe dreams are reduced, $\varphi$ produces a BPD with the same
weight, which is in bijection with an ASM of the same weight. Thus we have a
weight-preserving injection
$\varphi:\operatorname{TSSCPP}^{red}(\pi)\hookrightarrow\operatorname{ASM}^{red}(\pi)$
given by transfoming the TSSCPP to its corresponding reduced pipe dream as in
Theorem 3.3, mapping it to a reduced BPD using $\varphi$, and then
transforming to an ASM using the bijection described in Section 2.4.

Suppose $\pi$ avoids $1432$. Then by Lemma 4.3, all pipe dreams in
$\operatorname{PD}^{red}(\pi)$ are pseudo-Yamanouchi, so
$\operatorname{TSSCPP}^{red}(\pi)$ is in bijection with
$\operatorname{PD}^{red}(\pi)$. So the above injection is a bijection between
$\operatorname{TSSCPP}^{red}(\pi)$ and $\operatorname{ASM}^{red}(\pi)$. ∎

##  5\. A poset-preserving bijection in the $2143$\- and $1432$-avoiding case

In this section, we study posets constructed using simple slides on pipe
dreams and simple droops on bumpless pipe dreams. We use this understanding to
construct a bijection between $\operatorname{TSSCPP}^{red}(\pi)$ and
$\operatorname{ASM}(\pi)$ that preserves their poset structure in the case
that $\pi$ avoids both $1432$ and $2143$. Note that this bijection is not, in
general, the same as the bijection $\varphi$ used in Theorem 1.1; see Remark
5.11. Note also that when $\pi$ avoids $2143$,
$\operatorname{ASM}^{red}(\pi)=\operatorname{ASM}(\pi)$, since all BPD that
avoid $2143$ are reduced; see Lemma 5.3 below.

We use simple droops to define a poset on BPD. The Rothe BPD of $\pi$ is the
BPD corresponding to the permutation matrix of $\pi$. We will also refer to
the Rothe BPD as the Rothe diagram of $\pi$. Note that the connected regions
of blank tiles of a Rothe diagram are all partition-shaped.

######  Definition 5.1.

Given $\pi\in S_{n}$, let $\operatorname{\mathrm{Droop}}(\pi)$ denote the
poset constructed from applying simple droops in all possible ways to the
Rothe BPD of $\pi$. The Rothe BPD is the bottom element of the poset and each
simple droop moves up in the poset.

Note that this construction also induces a poset on the corresponding ASM, by
the simple bijection between BPD and ASM.

######  Definition 5.2.

The set of essential boxes of a permutation $\pi$ is the set
$\operatorname{ess}(\pi):=\\{(i,j):\pi(j)>i,\pi^{-1}(i)>j,\pi(j+1)\leq
i,\pi^{-1}(i+1)\leq j\\}$. In other words, $\operatorname{ess}(\pi)$ consists
of the SE-most corners in the connected regions of blank tiles of the Rothe
diagram of $\pi$. Define the dominant region of Rothe diagram to be the
connected region of blank tiles containing $(1,1)$. Note that the dominant
region might be empty.

The following statement is found in [Wei21].

######  Lemma 5.3 ([Wei21, Lemma 7.2, Lemma 7.4 (2)]).

If $\pi\in S_{n}$ avoids $2143$, then any $D\in\operatorname{BPD}(\pi)$ is
reduced, and any two bumpless pipe dreams of $\pi$ are connected by simple
droops.

This yields the following corollary.

######  Corollary 5.4.

If $\pi\in S_{n}$ avoids $2143$, the elements of
$\operatorname{\mathrm{Droop}}(\pi)$ are all of $\operatorname{BPD}(\pi)$.

We define a similar poset for pipe dreams, using the simple slides of Section
2.5.

######  Definition 5.5.

Given $\pi\in S_{n}$, let $\operatorname{\mathrm{Slide}}(\pi)$ denote the
poset constructed from applying simple slides to the bottom pipe dream in all
possible ways. The bottom pipe dream is the bottom element of this poset, and
each simple slide moves up in the poset.

We have the following, as a corollary of the result of Gao we stated as
Theorem 4.1.

######  Corollary 5.6.

If $\pi\in S_{n}$ avoids $1432$, the elements of
$\operatorname{\mathrm{Slide}}(\pi)$ are all of
$\operatorname{PD}^{red}(\pi)$.

An important class of permutations are the Grassmannian permutations, which
are defined as the permutations with at most one descent. Grassmannian
permutations necessarily avoid both $2143$ and $1432$, since these patterns
each have two descents. Inverse-Grassmannian permutations are permutations
whose inverse is Grassmannian. These also avoid both $2143$ and $1432$, since
these patterns are their own inverses.

We now relate $\operatorname{\mathrm{Droop}}(\pi)$ and
$\operatorname{\mathrm{Slide}}(\pi)$ in the cases that $\pi$ is inverse-
Grassmannian or Grassmannian. We will need the following lemmas, the first of
which is also due to Weigandt.

######  Lemma 5.7 ([Wei21, Lemma 7.4 (1)]).

If $\pi$ avoids $2143$, all bumpless pipe dreams in $\operatorname{BPD}(\pi)$
are uniquely determined by the locations of the blank tiles.

######  Lemma 5.8.

The blank tiles in the Rothe BPD of a Grassmannian permutation have the
following characterizing properties:

  1. (a)

Each connected region of blank tiles is a rectangular block.

  2. (b)

The NW-most blank tile of the first rectangular block is on the diagonal.

  3. (c)

The essential boxes lie in the same row.

  4. (d)

If $B_{1}$ and $B_{2}$ are two consecutive blocks and the horizontal distance
between them is $d$, then the height of $B_{1}$ is $d$ more than the width of
$B_{2}$.

Furthermore, if a set of blank tiles satisfy the properties above, this set
uniquely determines the Rothe BPD of a Grassmannian permutation. If we replace
(c) and (d) with the following

  1. (c′)

The essential boxes lie in the same column.

  2. (d′)

If $B_{1}$ and $B_{2}$ are two consecutive blocks and the vertical distance
between them is $d$, then the width of $B_{1}$ is $d$ more than the width of
$B_{2}$.

we get similar characterizing properties for the inverse-Grassmannian
permutations.

###### Proof.

We may consider the permutation as an element in $S_{\infty}$ and the BPDs to
be unbounded in the east and south directions. The Rothe BPD of an inverse
Grassmannian permutation is the transpose of that of a Grassmanian
permutation, so it suffices to argue for Grassmannian permutations. It is easy
to see the set of blank tiles of the Rothe BPD for a Grassmannian permutation
satisfies properties (a)–(d). Given a set of blank tiles satisfying (a)–(d),
we may construct a Rothe BPD by iteratively placing a  tile at the NW-most
undetermined position, and extending to the east by a horizontal ray and to
the south by a vertical ray. The properties (a)–(d) guarantee that this is
always possible, see Figure 6. ∎

Figure 6. The Rothe BPD of a Grassmannian permutation is determined by blank
tiles satisfying conditions (a)–(d) of Lemma 5.8. Transposing the diagrams we
get the analogous statements for inverse Grassmannians.

We are now ready to show a poset isomorphism between BPD and reduced PD in the
inverse-Grassmannian case.

######  Theorem 5.9.

If $\pi\in S_{n}$ is inverse-Grassmannian, there is an explicit weight-
preserving bijection between $\operatorname{BPD}(\pi)$ and
$\operatorname{PD}^{red}(\pi)$ such that $\operatorname{\mathrm{Droop}}(\pi)$
$\cong$ $\operatorname{\mathrm{Slide}}(\pi)$.

###### Proof.

Suppose $\pi\in S_{n}$ is inverse-Grassmannian. By Lemma 5.7, since $\pi$
avoids $2143$, the blank tiles of the BPD completely determine it. The map on
the Rothe $\operatorname{BPD}$ of $\pi$ that left-justifies all the blank
tiles and turns them into crosses results in the bottom (permutation) pipe
dream of $\pi$ [BB93]. By Lemma 5.8, the blank tiles in the Rothe BPD of $\pi$
are disconnected rectangular blocks that are aligned on the right. When they
are left-justified to create the bottom pipe dream, they become mirror-image
rectangular blocks aligned on the left. Furthermore, the NW-most blank tile in
the Rothe $\operatorname{BPD}$ is on the diagonal. The number of simple droops
that may be applied to this blank tile then equals the number of simple slides
that may be applied to the rightmost cross in this row of the bottom pipe
dream. Moreover, the simple droop moves of
$\operatorname{\mathrm{Droop}}(\pi)$ correspond exactly to the simple slides
of $\operatorname{\mathrm{Slide}}(\pi)$. Thus,
$\operatorname{\mathrm{Droop}}(\pi)$ $\cong$
$\operatorname{\mathrm{Slide}}(\pi)$. By construction, the number of blank
tiles in row $k$ of a BPD in $\operatorname{\mathrm{Droop}}(\pi)$ corresponds
to the number of cross tiles in row $k$ of the corresponding PD, so this
weight is preserved. See Figure 7 for an example. ∎

The following is a direct application of Theorem 5.9 to ASM and TSSCPP.

######  Corollary 5.10.

If $\pi\in S_{n}$ is inverse-Grassmannian, there is an explicit weight-
preserving bijection between $\operatorname{ASM}(\pi)$ and
$\operatorname{TSSCPP}^{red}(\pi)$ such that
$\operatorname{\mathrm{Droop}}(\pi)$ $\cong$
$\operatorname{\mathrm{Slide}}(\pi)$.

Figure 7. A weight-preserving and poset-preserving bijection between
$\operatorname{BPD}(14253)$ and $\operatorname{PD}(14253)$. Note that $14253$
is inverse-Grassmannian and avoids both $1432$ and $2143$.

######  Remark 5.11.

For crystal-theoretic reasons, the bijection of Corollary 5.10 coincides with
the bijection $\varphi$ of Theorem 1.1, though this will not be true for the
subsequent corollaries of this section.

We have a similar result for Grassmannian permutations, but the poset
isomorphism is with the dual poset $\operatorname{\mathrm{Slide}}(\pi)^{*}$
(all order relations reversed). Note that in the case that $\pi$ avoids 1432,
by Theorem 4.1, $\operatorname{\mathrm{Slide}}(\pi)$ has a unique maximal
element, the pipe dream in which all the crosses are top-justified; this is
called the top pipe dream. Thus, $\operatorname{\mathrm{Slide}}(\pi)^{*}$ has
the top pipe dream as its unique minimal element.

We now show a dual poset isomorphism between BPD and reduced PD in the
Grassmannian case. Note that the transpose of a Grassmannian permutation is
inverse-Grassmannian, so Grassmannian permutations satisfy the transpose of
the description in Theorem 5.9.

######  Theorem 5.12.

If $\pi\in S_{n}$ is Grassmannian, there is an explicit bijection between
$\operatorname{BPD}(\pi)$ and $\operatorname{PD}^{red}(\pi)$ such that
$\operatorname{\mathrm{Droop}}(\pi)$ $\cong$
$\operatorname{\mathrm{Slide}}(\pi)^{*}$.

###### Proof.

Suppose $\pi\in S_{n}$ is Grassmannian. By Lemma 5.7, since $\pi$ avoids
$2143$, the blank tiles of the BPD completely determine it. The map on the
Rothe $\operatorname{BPD}$ of $\pi$ that top-justifies all the blank tiles and
turns them into crosses results in the top pipe dream of $\pi$. This can be
seen by transposing the construction that left-justifies blank tiles of
inverse-Grassmannian Rothe BPDs as described in the proof of Theorem 5.9. By
Lemma 5.8, the blank tiles in the Rothe BPD of $\pi$ are disconnected
rectangular blocks that are aligned on the bottom. When they are top-justified
to create the top pipe dream, they become mirror-image rectangular blocks
aligned on the top. Furthermore, the NW-most blank tile in the Rothe
$\operatorname{BPD}$ is on the diagonal. The number of simple droops that may
be applied to this blank tile then equals the number of inverse simple slides
that may be applied to the lowest cross in this column of the top pipe dream.
Moreover, the simple droop moves of $\operatorname{\mathrm{Droop}}(\pi)$
correspond exactly to the inverse simple slides of
$\operatorname{\mathrm{Slide}}(\pi)$. Thus,
$\operatorname{\mathrm{Droop}}(\pi)$ $\cong$
$\operatorname{\mathrm{Slide}}(\pi)^{*}$. ∎

######  Remark 5.13.

Note that by construction of this poset isomorphism, the number of blank tiles
in row $k$ of a BPD in $\operatorname{\mathrm{Droop}}(\pi)$ corresponds to the
number of cross tiles row $n-k+1$ of the corresponding PD, so this weight is
reversed instead of preserved.

The corollary below follows immediately from Theorem 5.12.

######  Corollary 5.14.

If $\pi\in S_{n}$ is Grassmannian, there is an explicit bijection between
$\operatorname{ASM}(\pi)$ and $\operatorname{TSSCPP}^{red}(\pi)$ such that
$\operatorname{\mathrm{Droop}}(\pi)$ $\cong$
$\operatorname{\mathrm{Slide}}(\pi)^{*}$.

######  Remark 5.15.

One may wonder why we use the dual poset in this theorem/corollary when many
small examples, including that of Figure 7, seem to indicate that these posets
may be self-dual. An example of a Grassmannian permutation $\pi$ for which
$\operatorname{\mathrm{Slide}}(\pi)\neq\operatorname{\mathrm{Slide}}(\pi)^{*}$
is $\pi=146235$.

Now we generalize these correspondences to define a poset on
$\operatorname{PD}(\pi)$ in the case that $\pi$ avoids both $1432$ and $2143$
that is isomorphic to $\operatorname{\mathrm{Droop}}(\pi)$. This will yield a
poset-preserving bijection between the corresponding TSSCPP and ASM. First, we
characterize features of these BPD in the lemmas below.

The following statement is well-known. We include a short proof.

######  Lemma 5.16.

If $\pi$ is $2143$-avoiding, then essential boxes of the Rothe diagram of
$\pi$ run northeast to southwest. Namely, it is impossible that
$(i_{1},j_{1}),(i_{2},j_{2})\in\operatorname{ess}(\pi)$ such that
$i_{1}<i_{2}$ and $j_{1}<j_{2}$.

###### Proof.

Let $\pi$ be a permutation and suppose $(i_{1},j_{1})$ and $(i_{2},j_{2})$ are
two essential boxes of $\pi$. Suppose $i_{1}<i_{2}$ and $j_{1}<j_{2}$. Since
$(i_{1},j_{1})\in\operatorname{ess}(\pi)$, we have $\pi^{-1}(j_{1}+1)\leq
i_{1}$ and $\pi(i_{1}+1)\leq j_{1}$. Since
$(i_{2},j_{2})\in\operatorname{ess}(\pi)$, there exist $i_{1}+1<a\leq i_{2}$
such that $\pi(a)>j_{2}$ and $j_{1}+1<b\leq j_{2}$ such that
$\pi^{-1}(b)>i_{2}$. We then have $\pi^{-1}(j_{1}+1)<i_{1}+1<a<\pi^{-1}(b)$
where $\pi(i_{1}+1)<j_{1}+1<b<\pi(a)$, so $\pi$ must be 2143-containing. ∎

######  Lemma 5.17.

If $\pi$ is $1432$-avoiding, the Rothe diagram of $\pi$ satisfies the
following properties:

  1. (a)

The partition shapes formed by the blank region must all be rectangles, except
for the dominant region.

  2. (b)

Let $B_{1}$ and $B_{2}$ be two non-dominant connected regions of blank tiles
with essential boxes $(i_{1},j_{1})$ and $(i_{2},j_{2})$. If $B_{1}$ and
$B_{2}$ both contain tiles in some row $r$, then $i_{1}=i_{2}$. If $B_{1}$ and
$B_{2}$ both contain tiles in some column $j$, then $j_{1}=j_{2}.$

###### Proof.

For (a), suppose to the contrary that there is a non-rectangular partition
that is not dominant. Let $(a,b)$ be its NW corner, and $(i_{1},j_{1})$,
$(i_{2},j_{2})$ with $i_{1}<i_{2}$ and $j_{1}>j_{2}$ SE corners of this
partition with no other SE corners in between. Note that $a>1$. Let $(x,y)$ be
the unique outer NW corner SW of $(i_{1},j_{1})$ and NE of $(i_{2},j_{2})$. It
must be the case that $\pi(x)=y$. Then $\pi(a-1)<b$, $\pi(i_{1})>j_{1}$, and
$\pi^{-1}(j_{2})>i_{2}$. Therefore, $a-1<i_{1}<x<\pi^{-1}(j_{2})$ and
$\pi(a-1),\pi(i_{1}),\pi(x),j_{2}$ gives the pattern 1432, contradicting our
assumption on $\pi$.

For (b), we show the row statement and the column statement is similar. It
suffices to consider the case when $B_{1}$ and $B_{2}$ are adjacent and
$j_{1}<j_{2}$. Now suppose $i_{1}\neq i_{2}$. Since $B_{1}$ and $B_{2}$ both
contain tiles in some row $r$, we must have $i_{1}>i_{2}$. Let $(a,b)$ be the
NW corner of $B_{1}$, then $\pi(a-1)<b\leq j_{1}$. Since
$(i_{2},j_{2})\in\operatorname{ess}(\pi)$, $\pi(i_{2})>j_{2}$. Note also that
$j_{1}<\pi(i_{1})\leq j_{2}$. Finally, $\pi^{-1}(b)>i_{1}$. We then have
$a-1<i_{2}<i_{1}<\pi^{-1}(b)$ and $\pi(a-1)<b<\pi(i_{1})<\pi(i_{2})$ which
gives the pattern 1432. ∎

######  Lemma 5.18.

Suppose $\pi$ is $1432$\- and $2143$-avoiding. If
$(i,j_{1}),(i,j_{2})\in\operatorname{ess}(\pi)$, $j_{1}<j_{2}$ and $(i,j_{1})$
is not in the dominant region, then
$(i^{\prime},j_{1})\not\in\operatorname{ess}(\pi)$ for any $i^{\prime}>i$.
Similarly if $(i_{1},j),(i_{2},j)\in\operatorname{ess}(\pi)$, $i_{1}<i_{2}$
and $(i_{1},j)$ is not in the dominant region, then
$(i_{1},j^{\prime})\not\in\operatorname{ess}(\pi)$ for any $j^{\prime}>j$.

###### Proof.

Suppose $(i_{1},j_{1}),(i_{1},j_{2}),(i_{2},j_{1})\in\operatorname{ess}(\pi)$
with $i_{1}<i_{2}$ and $j_{1}<j_{2}$, and $(i_{1},j_{1})$ is not in the
dominant region. Suppose $B$, $B_{1}$, $B_{2}$ are the blank regions
containing $(i_{1},j_{1})$, $(i_{1},j_{2})$, and $(i_{2},j_{1})$,
respectively. Suppose $(a_{1},b)$ is the NW corner of $B$ and $a_{2}$ is the
row index of the topmost row in $B_{2}$. Then $\pi(a-1)<b\leq j_{1}$,
$\pi(i_{1})>j_{2}$, and $\pi^{-1}(j_{1})>i_{2}$. Since the essential boxes of
2143-avoiding permutations run NE to SW by Lemma 5.16, there can be no blank
tiles strictly SE of $(i_{1},j_{1})$, and therefore $j_{1}<\pi(a_{2})\leq
j_{2}$. We then have $a-1<i_{1}<a_{2}<\pi^{-1}(j_{1})$ and
$\pi(a-1)<j_{1}<\pi(a_{2})<\pi(i_{2})$ giving the pattern 1432. ∎

######  Definition-Lemma 5.19.

If $\pi\in S_{n}$ is $2143$\- and $1432$-avoiding, the Rothe
$\operatorname{BPD}$ can be partitioned into a dominant partition of empty
squares in the NW corner, a partition shape of fixed pipes in the SE corner,
and disjoint blocks containing non-intersecting pipes corresponding to
Grassmannian and inverse-Grassmannian permutations. Call this the block-
decomposition of $\pi$.

###### Proof.

By Lemmas 5.17 and 5.18, the set of essential boxes outside of the dominant
region can be partitioned into $E_{1},\cdots,E_{m}$ according to whether they
lie on the same row or column. In particular, if $e_{1},e_{2}\in E_{i}$ for
some $i$, then either $e_{1}$ and $e_{2}$ are in the same row, or they are in
the same column. If $E_{i}$ contain essential boxes in the same row $r$, let
$c$ be the smallest column index such that $(r,c)$ is not in the dominant
region. Let $r^{\prime}<r$ be the smallest row index such that
$(r^{\prime},c)$ is not in the dominant region. Let $l$ be the sum of the
widths of the rectangular blocks of blank tiles with essential boxes in
$E_{i}$. Then the rectangular region with $(r,c)$ as the SW corner of height
$r-r^{\prime}+1$ and width $l+r-r^{\prime}+1$ consist of $r-r^{\prime}+1$ non-
intersecting pipes and agree with the top $r-r^{\prime}+1$ rows of the Rothe
BPD of a Grassmannian permutation by Lemma 5.8. Transposing this construction
we get a rectangular region of inverse-Grassmannian permutation. The rest of
the Rothe diagram contains fixed pipes, as there are no more blank tiles. ∎

######  Theorem 5.20.

Suppose $\pi\in S_{n}$ avoids both $2143$ and $1432$. Let
$\sigma_{1},\sigma_{2},\ldots,\sigma_{k}$ be the inverse-Grassmannian
permutations in the block-decomposition of $\pi$ and
$\tau_{1},\tau_{2},\ldots,\tau_{\ell}$ the Grassmannian permutations in the
block-decomposition of $\pi$. There is an explicit bijection between
$\operatorname{BPD}(\pi)$ and $\operatorname{PD}^{red}(\pi)$ such that

$\operatorname{\mathrm{Droop}}(\pi)\cong\operatorname{\mathrm{Slide}}(\sigma_{1})\times\operatorname{\mathrm{Slide}}(\sigma_{2})\times\cdots\times\operatorname{\mathrm{Slide}}(\sigma_{k})\times\operatorname{\mathrm{Slide}}(\tau_{1})^{*}\times\operatorname{\mathrm{Slide}}(\tau_{2})^{*}\times\cdots\times\operatorname{\mathrm{Slide}}(\tau_{\ell})^{*}.$

###### Proof.

By Lemma 5.19, the Rothe diagram of such a permutation is block-Grassmannian
and inverse-Grassmannian. So we begin by mapping the Rothe diagram to a PD
block-by-block. For the Grassmannian blocks, we map to the top pipe dream, and
for the inverse-Grassmannian blocks, we map to the bottom pipe dream. Then
from Theorems 5.9 and 5.12, the posets for each block are isomorphic or dual
isomorphic. So we have the stated poset isomorphism. ∎

The corollary below follows immediately.

######  Corollary 5.21.

If $\pi\in S_{n}$ avoids both $2143$ and $1432$, there is an explicit
bijection $\psi$ between $\operatorname{ASM}(\pi)$ and
$\operatorname{TSSCPP}^{red}(\pi)$ such that

$\operatorname{\mathrm{Droop}}(\pi)\cong\operatorname{\mathrm{Slide}}(\sigma_{1})\times\operatorname{\mathrm{Slide}}(\sigma_{2})\times\cdots\times\operatorname{\mathrm{Slide}}(\sigma_{k})\times\operatorname{\mathrm{Slide}}(\tau_{1})^{*}\times\operatorname{\mathrm{Slide}}(\tau_{2})^{*}\times\cdots\times\operatorname{\mathrm{Slide}}(\tau_{\ell})^{*}$

where $\sigma_{1},\sigma_{2},\ldots,\sigma_{k}$ are the inverse-Grassmannian
permutations and $\tau_{1},\tau_{2},\ldots,\tau_{\ell}$ the Grassmannian
permutations in the block-decomposition of $\pi$.

######  Example 5.22.

In Figure 8, we show an example of the Rothe BPD for a 2143- and 1432-
avoiding permutation $\pi$ and its image under the poset-preserving bijection
$\psi$. The dominant partition in the upper left corner of each diagram is
$(6,6,6,5,4)$. The block-decomposition of $\pi$ (from Definition-Lemma 5.19)
consists of the Grassmannian permutation $\sigma=146235$ (shown in the upper
right of each diagram) and the inverse-Grassmannian permutation $\tau=142563$
(shown in the lower left of each diagram).

Figure 8. Example of an Rothe BPD for a 2143- and 1432- avoiding permutation
and its image under the poset-preserving bijection $\psi$

##  6\. Concluding remarks

We conclude the paper by discussing the progress made thus far on the ASM-
TSSCPP bijection problem and an outlook on and challenges to further progress.
Table 1 below shows the numbers up to $n=7$ (computed using `SageMath` [S+22])
of ASM and TSSCPP that correspond using the theorems of this paper (Columns 3,
6, 7) as well as the results of [Str18, ACGB11, Str11, CB12, Bet19] (Columns
2, 4, 5).

  * •

Column 2 is given by $n!$, as this counts the permutation case bijection of
[Str18].

  * •

Column 3 gives those matched under the poset-preserving $1432$\- and
$2143$-avoiding bijection of the present paper (Corollary 5.21).

  * •

Column 4 gives the number of ASM and TSSCPP matched in the bijection of
[ACGB11, Theorem 4], which concerns the case that the monotone triangle
associated to the ASM is _gapless_ ; this is the same set as the intersection
discussed in [Str11, Section 7].

  * •

Column 5 gives the number matched in the bijections of [CB12, Bet19], which
both concern the case of at most two nontrivial diagonals in the monotone
triangle.

  * •

Column 6 gives the number matched in the weight-preserving $1432$-avoiding
bijection of the present paper (Theorem 1.1).

  * •

Column 7 gives the number matched in the weight-preserving injection of
Theorem 1.1 on all TSSCPP whose pipe dreams are reduced.

  * •

Column 8 gives the total number of ASM, for comparison.

One may ask whether any of these bijections include any of the other partial
bijections. As noted earlier, the weight-preserving injection of Theorem 1.1
(Column $7$) extends the permutation case bijection of [Str18] (Column $2$).
Also, the subsets addressed in this paper are proper subsets of each other
(Columns 3, 6, 7). But it is useful to note that the subset of ASM included in
the other bijections discussed here (Columns 4 and 5) is not a proper subset
of the ASM included in Theorem 1.1. In particular, the one ASM for $n=4$ whose
BPD is non-reduced has a monotone triangle that is both gapless and having
only two non-trivial diagonals. Thus, it is included in the bijections of
Columns 4 [ACGB11, Str11] and 5 [CB12, Bet19], but not in Theorem 1.1 (Column
7).

Pattern avoidance is discussed in [ACGB11], in the sence that the bijection
$\Delta$ of [ACGB11, Theorem 4] includes all permutations that avoid the
pattern $312$. But this notion of pattern avoidance does not coincide with
ours outside of permutations. For instance, $\Delta$ does not include the
entire set $\operatorname{ASM}(2143)$, even though $2143$ avoids $312$, but it
does include an alternating sign matrix in $\operatorname{ASM}(3142)$, even
though $3142$ contains the pattern $312$.

Size | Perm | (1432,2143)-av | Gapless | Two-diag | 1432-av | Matched in | Total num
---|---|---|---|---|---|---|---
| bijection | bijection | bijection | bijection | bijection | injection | of ASM
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
2 | 2 | 2 | 2 | 2 | 2 | 2 | 2
3 | 6 | 7 | 6 | 7 | 7 | 7 | 7
4 | 24 | 33 | 26 | 35 | 36 | 40 | 42
5 | 120 | 185 | 162 | 219 | 246 | 362 | 429
6 | 720 | 1175 | 1450 | 1594 | 2135 | 5125 | 7436
7 | 5040 | 8261 | 18626 | 12935 | 23067 | 112941 | 218348
Table 1. The number of ASM and TSSCPP in correspondence via the various
results of this paper and as compared to other subset bijections. The column
headings in bold represent results from this paper.

One may ask whether it is possible to extend the bijection of Theorem 1.1
beyond $1432$-avoiding permutations and/or remove the reducedness
restrictions. There are some challenges. In the case of $n=4$, $40$ pseudo-
Yamanouchi pipe dreams are reduced, so all the corresponding TSSCPP are mapped
to reduced BPD, and therefore to ASM. There are only two remaining TSSCPPs,
shown in the left column of Figure 9. There is one remaining reduced pipe
dream, shown in the bottom middle of Figure 9, which maps by $\varphi$ to the
BPD on the bottom right. Comparing this reduced pipe dream with the two
remaining TSSCPP pipe dreams, we see it differs from the one with three
crosses by moving the top cross to the right, creating a non-reduced pipe
dream. In general, the set of reduced pipe dreams for a fixed permutation are
connected by $(n\times 2)$–ladder moves and $(2\times n)$–chute moves [BB93],
and these moves do not always preserve the pseudo-Yamanouchi property when
$n>2$.

In the forthcoming work of Shimozono, Yu, and the first author, a weight-
preserving bijection between the set of all $2^{\binom{n}{2}}$ pipe dreams in
$\operatorname{PD}(n)$ and the set of all marked bumpless pipe dreams (a
marked BPD is a BPD whose -tiles admit a binary marking) of size $n$, which
generalizes the bijection in [GH21]. Under this bijection, the BPD on the top
right of Figure 9 is mapped to the PD on the top middle. Notice that the one
remaining TSSCPP PD on the top left has one more , so a bijection that
preserves this weight is no longer possible.

Figure 9. Challenges for the remaining unmatched TSSCPP pipe dreams and
bumpless pipe dreams

## Acknowledgments

The authors would like to thank Yibo Gao for helpful comments. They also thank
the developers of `SageMath` [S+22] software, which was useful in this
research. They thank O. Cheong for developing Ipe [Che20], which we used to
create the figures in this paper. JS was supported by a grant from the Simons
Foundation/SFARI (527204, JS). DH was supported by NSF-DMS2202900.

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