# Twenty Vertex model and domino tilings of the Aztec triangle

Philippe Di Francesco  Department of Mathematics, University of Illinois,
Urbana, IL 61821, U.S.A. and Institut de physique théorique, Université Paris
Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, FRANCE e-mail:
<EMAIL_ADDRESS>
###### Abstract.

We show that the number of configurations of the 20 Vertex model on certain
domains with domain wall type boundary conditions is equal to the number of
domino tilings of Aztec-like triangles, proving a conjecture from [DFG20]. The
result is based on the integrability of the 20 Vertex model and uses a
connection to the U-turn boundary 6 Vertex model to re-express the number of
20 Vertex configurations as a simple determinant, which is then related to a
Lindström-Gessel-Viennot determinant for the domino tiling problem. The common
number of configurations is conjectured to be
$2^{n(n-1)/2}\prod_{j=0}^{n-1}\frac{(4j+2)!}{(n+2j+1)!}=1,4,60,3328,678912...$
The enumeration result is extended to include refinements of both numbers.

###### Contents

  1. 1 Introduction
    1. 1.1 Triangular Ice model combinatorics
    2. 1.2 Outline of the paper and main results
  2. 2 Partition function of the 20V model
    1. 2.1 The 20V model: integrable weights
    2. 2.2 Boundary conditions
    3. 2.3 Transformation to a mixed 6V model
    4. 2.4 Transformation to a U-turn boundary 6V model
    5. 2.5 Inhomogeneous determinant formula
    6. 2.6 Homogeneous limit
    7. 2.7 The 20V model partition function
  3. 3 Domino tilings of the Aztec triangle
    1. 3.1 Path model
    2. 3.2 Generating functions and determinant formula
  4. 4 20V-Domino Tiling correspondence
    1. 4.1 Determinant relations
    2. 4.2 Proof of the equivalence theorem
  5. 5 Refined partition functions of the 20V model
    1. 5.1 Setting and weights
    2. 5.2 Refined partition functions
    3. 5.3 Transformations to the 6V model
    4. 5.4 Refined partition functions of the 6V model
    5. 5.5 Relation to the inhomogeneous U-turn boundary partition function
    6. 5.6 Refined U-turn boundary
    7. 5.7 Refined partition functions for 20V and 6V models
  6. 6 Refined domino tilings of the Aztec triangle and connection to 20V refined partition functions
    1. 6.1 Refined domino tiling partition functions
    2. 6.2 Relation between the refined tiling and refined 20V partition functions
  7. 7 Pentagonal 20V and truncated Aztec triangle domino tilings
    1. 7.1 Proof of the 20V-DT conjecture for $k=2$
    2. 7.2 Proof of the 20V-DT conjecture for $k=3$
  8. 8 Discussion and Conclusion
    1. 8.1 An exact (conjectured) formula
    2. 8.2 Alternative expressions for $Z_{n}^{20V}$
    3. 8.3 Introducing step weights in the domino tiling problem
    4. 8.4 Conclusion

##  1\. Introduction

###  1.1. Triangular Ice model combinatorics

Figure 1. The local vertex configurations of the 20V model (top) and their
reformulation in terms of osculating Schröder paths (bottom).

Two-dimensional integrable lattice models of statistical physics have
exhibited an extremely rich mathematical content, ranging from representation
theory to probability and combinatorics. The present paper concentrates on the
two-dimensional triangular lattice111Throughout the paper we view the two-
dimensional triangular lattice in a sheared fashion: the vertices form a
square lattice, and edges are those of the square lattice plus the second
diagonal within each square. This allows for easier drawings, and is more
adapted to our choices of domains. version of the ice-type models, in the form
of the so-called Twenty Vertex (20V) model [Kel74, Bax89]. The model is
defined on a finite domain of the triangular lattice, by considering all
possible orientations of edges of the elementary triangles, obeying the ice
rule: “there are equal numbers of incoming and outgoing edges at each vertex
of the domain”. This gives rise to the ${6\choose 3}=20$ local vertex
configurations displayed in the top two rows of Fig. 1. The 20V model is
really the triangular lattice version of the celebrated Six Vertex (6V) model
[Lie67] describing ice on a two-dimensional square lattice. The latter was at
the core of the saga of Alternating Sign Matrices (ASM) [Bre99], a remarkable
sequence of connections between purely combinatorial objects such as
Descending Plane Partitions (DPP) [And80] in bijection with cyclically
symmetric rhombus tilings of an hexagon with a central triangular hole
[Kra06], Totally Symmetric Self-Complementary Plane Partitions (TSSCPP)
[MRR86, Zei96] and two-dimensional models of statistical physics such as the
6V model [ICK92, Kup96, Tsu98, Kup02], the O(1) loop model [MNdGB04] and
finally the Fully-Packed Loop model (FPL), at the center of the Razumov-
Stroganov conjecture [RS04a], later proved by Cantini and Sportiello [CS11]. A
striking feature of all the correspondences is the apparent absence of natural
bijections between different classes of objects with the same cardinality
$A_{n}=\prod_{j=0}^{n-1}\frac{(3j+1)!}{(n+j)!}=1,2,7,42,429,...$ The cases
studied in this paper can be seen as other remarkable examples of coincidences
of cardinalities of sets, between which no natural bijection is known.

Figure 2. DWBC1 boundary conditions for the 20V model and a sample
configuration in the osculating Schröder path formulation.

In its physics literature debut, the 20V model was shown to be integrable
provided some suitable choice of Boltzmann weights is made [Kel74, Bax89],
governed by 3 independent parameters, for which the thermodynamic free energy
was computed. This used an explicit connection between the integrable 20V
model on the triangular lattice and the integrable 6V model on the square
lattice. Recently the model was revisited from a combinatorial perspective
[DFG20], by considering domains with so-called “domain-wall boundary
conditions” (DWBC), i.e. by imposing orientations on the boundaries of the
domain that force the existence of non-local fault lines within the domain,
across which orientations are reversed (domain walls). In [DFG20], several
possible DWBC were considered (labeled 1,2,3,4) on a square $n\times n$ grid.
Each type of DWBC gives rise to generalizations of ASM, coined Alternating
Phase Matrices with entries among $0$ and the sixth roots of unity and
conservation conditions along rows, columns and diagonal lines.

The 20V model can be reformulated in terms of lattice paths, by picking a
preferred orientation of all edges of the lattice (say right, down and
diagonal down), and deciding that in any given orientation configuration, only
edges with the preferred orientation are path steps. There are 3 possibilities
for these steps: $(1,0),(0,-1),(1,-1)$ namely right, down and diagonal down.
They form non-intersecting but possibly “kissing” paths (called osculating
Schröder paths) that are allowed to share vertices, where up to three paths
bounce against each-other. The 20 local configurations of osculating Schröder
paths are displayed in the two bottom rows of Fig. 1.

In the case of DWBC1,2 illustrated in Fig. 2222Ref. [DFG20] considers two
choices labelled 1,2 for this first DWBC, which turn out to be equivalent
modulo a $180^{\circ}$ rotation, and we show here a DWBC1 example., it was
shown in [DFG20] that the partition function of the corresponding 20V model
with uniform weights counts also the number of $2\times 1$ cyclically
symmetric Domino Tilings of a Holey Aztec Square (HASDT), i.e. tilings of an
Aztec-like square shaped domain of the square lattice with a central cross-
shaped hole and with quarter-turn symmetry: this correspondence can be viewed
as a generalization of the ASM-DPP correspondence [BDFZJ12]. The proof relies
on the integrability of the 20V model allowing to transform the structure of
the underlying lattice by moving lines around, and eventually relating the 20V
partition function to that of the 6V model on a square grid with DWBC. As
opposed to the case of ASM, which correspond to the 6V model with quantum
parameter $q=e^{2i\frac{\pi}{3}}$, the 20V DWBC1,2 APM are enumerated by the
partition function of a 6V DWBC model with $q=e^{i\frac{\pi}{8}}$. The
connection between 20V and 6V being still valid in the presence of some non-
trivial spectral parameters, the result was generalized to include refinements
as well.

Figure 3. Extended DWBC3 boundary conditions on the pentagon
${\mathcal{P}}_{n,k}$ for the 20V model and a sample configuration in the
osculating Schröder path formulation, with trivial forced vertices removed.
Figure 4. Extended Pachter’s triangular domains ${\mathcal{T}}_{n,k}$ for
$n=4$ (bottom row), and the corresponding $20V$ pentagonal domains
${\mathcal{P}}_{n,k}$ (top row). The domain ${\mathcal{T}}_{n,k}$ must be
tiled by means of $2\times 1$ dominos, whereas ${\mathcal{P}}_{n,k}$ must
carry configurations of the $20V$ model with the specified boundaries.

In the same paper, other combinatorial conjectures were made regarding the 20V
DWBC3 model on a more general pentagonal grid ${\mathcal{P}}_{n,k}$, $0\leq
k\leq n-1$, illustrated in Fig. 3. It was conjectured in [DFG20] that the 20V
DWBC3 configurations on ${\mathcal{P}}_{n,k}$ are equinumerous to the $2\times
1$ domino tiling configurations of extensions $\mathcal{T}_{n,k}$ of the
triangular domain $\mathcal{T}_{n,0}$ of the square lattice introduced by
Pachter [Pac97] (see Fig. 4 for an illustration). In particular, for $k=0$ and
$n=1,2,...$ the number of configurations of the 20V DWBC3 model on the square
grid ${\mathcal{P}}_{n,0}$ forms the sequence $1,3,29,901,89893...$ which
matches conjecturally that of domino tilings of the Pachter triangle
$\mathcal{T}_{n,0}$.

The aim of the present paper is to shed some light on these latter conjectures
and to prove some of them. More precisely, we will concentrate on the 20V
DWBC3 model on the maximal domains $\mathcal{Q}_{n}:=\mathcal{P}_{n,n-1}$ (as
in Fig. 4 top right), whose configuration numbers give rise to the sequence
$1,4,60,3328,678912...$ and prove that they are counted by the numbers of
$2\times 1$ domino tilings of the maximally extended Aztec triangle
$\mathcal{T}_{n}:=\mathcal{T}_{n,n-1}$ (as in Fig. 4 bottom right) .

###  1.2. Outline of the paper and main results

The paper is organized as follows.

The partition function of the 20V model is computed in Section 2. Using
integrability, we transform the 20V model into a 6V model on a rectangular
grid with different boundary conditions, the so-called U-turn boundary
conditions333Remarkably, the same model but with the quantum parameter
$q=e^{2i\frac{\pi}{3}}$ was shown to enumerate Vertically Symmetric ASM
(VSASM) [Kup02]. [Kup02], and with quantum parameter $q=e^{i\frac{\pi}{8}}$.
The fully inhomogeneous version of the 20V model involves a large number of
arbitrary “spectral” parameters, which can be specialized to the combinatorial
point where all the Boltzmann weights are $1$. On the other hand, the 6V
U-turn boundary condition partition function is expressed as an explicit
determinant involving the spectral parameters of the model (Section 2.5). The
main difficulty in computing the 20V model partition function is to derive the
homogeneous limit of this determinant, when all spectral parameters tend to
their combinatorial point values. The result takes the form of a compact
determinant expression for the total number $Z_{n}^{20V}$ of configurations of
the 20V model on the domain $\mathcal{Q}_{n}$ (Theorem 2.7):

######  Theorem 1.1.

The total number $Z_{n}^{20V}$ of configurations of the 20V DWBC3 model on the
domain $\mathcal{Q}_{n}$ reads:

$Z_{n}^{20V}=\det_{0\leq i,j\leq
n-1}\left(\left.\frac{(1+u^{2})(1+2u-u^{2})}{(1-u^{2}v)\Big{(}(1-u)^{2}-v(1+u)^{2}\Big{)}}\right|_{u^{i}v^{j}}\right)$

The enumeration of domino tilings $Z_{n}^{DT}$ of the Aztec triangle
$\mathcal{T}_{n}$ is performed in Section 3 by a bijective formulation in
terms of non-intersecting Schröder lattice paths, leading to a compact
determinant formula (Theorem 3.4):

######  Theorem 1.2.

The total number $Z^{DT}_{n}$ of domino tilings of the Aztec triangle
$\mathcal{T}_{n}$ reads:

$Z^{DT}_{n}=\det_{0\leq i,j\leq
n-1}\left(\left.\frac{1+u}{1-v-4uv-u^{2}v+u^{2}v^{2}}\right|_{u^{i}v^{j}}\right)$

The identity between 20V and domino tiling enumerations is proved in Section 4
by noting that both enumeration results of Theorems 2.7 and 3.4 express the
numbers as the determinant of a finite truncation of infinite matrices $P$ and
$M$ respectively, and by exhibiting an explicit infinite lower triangular
matrix $L$ with entries 1 on the diagonal such that $L\,M=P$ (Theorem 4.3):

######  Theorem 1.3.

The partition function of the 20V DWBC3 model on the quadrangle
${\mathcal{Q}}_{n}$ and that of the domino tilings of the Aztec triangle
$\mathcal{T}_{n}$ coincide for all $n\geq 1$.

The second part of the paper is devoted to a refinement of the enumeration
result, in which we keep track of one particular statistic on the set of
configurations. For 20V configurations, we record the position of the first
visit to the rightmost vertical column. For domino tilings, we record the
first visit to the line of slope $-1$ through the topmost endpoint of the
paths. The relation between the two refinements is obtained by repeating the
analysis of the previous sections.

In Section 5, we follow the transformation of the 20V model into the U-turn 6V
model while keeping an arbitrary spectral parameter $w$ in the last column,
all other parameters being sent to their combinatorial point values. This
gives rise to a compact determinant formula for the generating polynomial of
refined 20V partition functions in the variable
$\tau=\frac{q^{-2}-q^{2}w}{q^{2}-q^{-2}w}$ (Theorems 5.3 and 5.4).

In Section 6, we use again the non-intersecting Schröder lattice path
formulation of the domino tiling problem to derive a compact determinant
formula for a generating polynomial of the refined partition functions
(Theorem 6.1).

In Section 6.2, the results of Theorems 5.3 and 6.1 are shown to be equivalent
(Theorem 6.2). The proof, inspired by techniques employed in [BDFZJ12, DFL18]
relies on the observation that both results are perturbations of the uniform
case, in which both infinite matrices $M,P$ are modified into infinite
matrices ${\bar{M}}^{(n)}(\tau),P^{(n)}(\tau)$ by an additional term affecting
only their $n$-th and higher columns, and for which the relation
$L\,{\bar{M}}^{(n)}(\tau)=P^{(n)}(\tau)$ still holds.

Section 7 is devoted to an investigation of the other conjectures regarding
the 20V model on $\mathcal{P}_{n,n-k}$ and the domino tilings of
$\mathcal{T}_{n,n-k}$ for $k>1$. We present direct proofs of the conjectures
for $k=2$ and $k=3$ based on recursion relations involving the $k=1$ result
and its refinements.

A few concluding remarks are gathered in Section 8. In Section 8.1, we present
the following:

######  Conjecture 1.4.

The total numbers $Z^{20V}_{n}=Z_{n}^{DT}$ of configurations of the 20V DWBC3
model on the quadrangle $\mathcal{Q}_{n}$ and of domino tilings of the Aztec
triangle $\mathcal{T}_{n}$ read:

(1.1)
$Z^{20V}_{n}=Z_{n}^{DT}=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!}$

for which we have no proof at this time. In Section 8.2, we present
alternative formulas for the numbers $Z^{20V}_{n}=Z_{n}^{DT}$, including a
determinant involving binomials (Theorem 8.2) and a constant term formula
(Theorem 8.4), in the same spirit as the TSSCP enumeration of Zeilberger
[Zei96]. In Section 8.3, we show how a simple modification of the results of
this paper allows to include an extra weight $\gamma$ per horizontal step in
the non-intersecting Schröder lattice path formulation of the domino tiling
problem. Section 8.4 is the conclusion per se.

Acknowledgments. We acknowledge partial support from the Morris and Gertrude
Fine endowment and the NSF grant DMS18-02044.

##  2\. Partition function of the 20V model

###  2.1. The 20V model: integrable weights

Given a domain $D$ of the two-dimensional triangular lattice, we consider the
set of orientations of all lattice edges within $D$ that satisfy the ice rule,
namely that each vertex is incident to exactly 3 incoming and 3 outgoing
edges. Boundary edge orientations are fixed, as part of the definition of the
model. As mentioned before, the 20V model configurations are alternatively
described by osculating Schröder paths on edges of the domain $D$ that travel
right and down. The boundary edges of $D$ are fixed to be occupied/empty as
part of the definition of the model. The occupied boundary edges split into
equal numbers of starting and endpoints of the paths.

Figure 5. The local vertex configurations of the 6V model (top row) and the
corresponding osculating path configurations (bottom row). We have indicated
the weights $a,b,c$ of the respective configurations.

Let us now describe the weighting of the configurations. Each vertex receives
a local weight according to its configuration, and each configuration on $D$
receives a weight equal to the product of the local weights of its inner
vertices. The weights are chosen in a special manner, which we describe now.
We attach complex “spectral parameters” to the lines of the lattice that
intersect $D$. If a vertex is at the intersection of three such lines, say
with parameters $z,t,w$, we attach a weight $W_{i}(z,t,w)$, $i=1,2...,20$
according to the list in Fig. 1. In [Kel74, Bax89, DFG20], the weights are
built out of elementary pieces as follows.

Consider a triple intersection between a horizontal, vertical and diagonal
line, and translate slightly the diagonal, say to the right: . This amounts to
resolving the triple intersection into three simple intersections, labeled
$1,2,3$.

Figure 6. The local vertex configurations of the three 6V models after
resolving all triple intersections in the 20V model.

Assume the edges were originally oriented and obeying the ice rule. The
resolution above keeps the six outer edge orientations and creates 3 new inner
edges, which may be oriented in such a way that the ice rule is now satisfied
at each simple intersection, which now has 4 adjacent edges. At each simple
intersection, this gives rise to ${4\choose 2}=6$ possible local
configurations, those of the so-called six vertex (6V) model (see Fig. 5 top
for a list of the vertex environments and their 3 customary weights $a,b,c$
that are invariant under reversal of all orientations). Here we are really
dealing with three different 6V models, attached to the three simple
intersections. The 6V weights are parameterized by the two spectral parameters
of their two lines, and an additional index 1,2,3 corresponding to
respectively a horizontal-vertical, horizontal-diagonal, and diagonal-vertical
intersection. The three sets of 6V weights are denoted by $a_{i},b_{i},c_{i}$
and are functions of their two spectral parameters (see Fig. 6 for the list of
all 18 possible configurations and their weights). The 20V weights can now be
defined in terms of the 6V weights of the resolution as follows:

$W^{20V}(z,t,w):=\sum_{{\rm inner}\ {\rm edge}\ {\rm
configs}.}w^{6V}_{1}(z,w)\,w^{6V}_{2}(z,t)\,w^{6V}_{3}(t,w)$

where the sum extends over all allowed inner edge orientations, and
$w^{6V}_{i}\in\\{a_{i},b_{i},c_{i}\\}$ stands for the 6V weight of the
corresponding local configuration. For instance, the resolution:

(2.1)

corresponds a 20V weight $\omega=a_{1}b_{2}c_{3}+c_{1}c_{2}b_{3}$.

The weights are further constrained by the integrability condition which
amounts to requiring that these values be independent of the resolution. For
instance, we could have moved the diagonal slightly to the left instead of the
right, giving rise to an a priori different definition of the $W^{20V}$s. The
requirement that the two definitions coincide results in a system of cubic
relations for the 6V weights (the celebrated Yang-Baxter equations). In the
case (2.1), the Yang-Baxter equation reads
$\omega=a_{1}b_{2}c_{3}+c_{1}c_{2}b_{3}=b_{1}a_{2}c_{3}$. We use the solution
of these equations from [DFG20], equivalent to that of [Kel74, Bax89]. Define
the following three functions of the two spectral parameters $z,w$:

(2.2) $A(z,w)=z-w,\quad B(z,w)=q^{-2}z-q^{2}w,\quad
C(z,w)=(q^{2}-q^{-2})\sqrt{zw},$

and pick the following 6V weights:

(2.3)
$\small{\begin{matrix}a_{1}=A(z,w)=z-w,\hfill&b_{1}=B(z,w)=q^{-2}z-q^{2}w,\hfill&c_{1}=C(z,w)=(q^{2}-q^{-2})\sqrt{zw}\hfill\\\
a_{2}=A(qz,q^{-1}t)=qz-q^{-1}t,\hfill&b_{2}=B(qz,q^{-1}t)=q^{-1}z-qt,\hfill&c_{2}=C(qz,q^{-1}t)=(q^{2}-q^{-2})\sqrt{zt}\hfill\\\
a_{3}=A(qt,q^{-1}w)=qt-q^{-1}w,\hfill&b_{3}=B(qt,q^{-1}w)=q^{-1}t-qw,\hfill&c_{3}=C(qt,q^{-1}w)=(q^{2}-q^{-2})\sqrt{tw}\hfill\end{matrix}}$

Figure 7. Integrable weights of the 20V model and their expression in terms of
weights of the three sublattice 6V models.

It is easy to show that the Yang-Baxter equations are satisfied by these, and
lead to the following expressions for the 20V integrable weights:

$\displaystyle\omega_{0}$ $\displaystyle=$
$\displaystyle(z-w)(qz-q^{-1}t)(qt-q^{-1}w)$ $\displaystyle\omega_{1}$
$\displaystyle=$ $\displaystyle(q^{-2}z-q^{2}w)(qz-q^{-1}t)(q^{-1}t-qw)$
$\displaystyle\omega_{2}$ $\displaystyle=$
$\displaystyle(q^{-2}z-q^{2}w)(qz-q^{-1}t)(q^{2}-q^{-2})\sqrt{tw}$
$\displaystyle\omega_{3}$ $\displaystyle=$ $\displaystyle
ztw(q^{2}-q^{-2})^{3}+(z-w)(q^{-1}z-qt)(q^{-1}t-qw)$ $\displaystyle\omega_{4}$
$\displaystyle=$
$\displaystyle(q^{2}-q^{-2})\sqrt{zw}(qz-q^{-1}t)(qt-q^{-1}w)$
$\displaystyle\omega_{5}$ $\displaystyle=$
$\displaystyle(q^{-2}z-q^{2}w)(q^{2}-q^{-2})\sqrt{zt}(qt-q^{-1}w)$ (2.4)
$\displaystyle\omega_{6}$ $\displaystyle=$
$\displaystyle(q^{-2}z-q^{2}w)(q^{-1}z-qt)(qt-q^{-1}w)$

where the index refers to the vertex configurations of Fig. 7 (our running
example (2.1) corresponds to $\omega=\omega_{2}$).

We will be first interested in the pure enumeration of the configurations of
the 20V model on specific domains with prescribed boundary conditions, which
requires uniform weights for all vertices. As it turns out, there exists a
choice of the spectral parameters and of the parameter $q$ which ensures that
all the weights (2.1) are equal to $1$.

######  Definition 2.1.

The “combinatorial point” where all $\omega_{i}=1$ corresponds to the values

(2.5) $q=e^{i\pi/8}\quad{\rm
and}\quad(z,t,w)=\alpha(q^{6},1,q^{-6}),\quad\alpha=2^{-5/6}q^{-4}$

The integrability property of the weights (2.1) will be crucially used in this
paper to transform partition functions. The property can be used to
deform/move any line of the lattice without changing the value of quantities
such as the partition function, i.e. the weighted sum over configurations. For
that, the presence of arbitrary spectral parameters is essential.
Combinatorial (enumeration) results will be obtained by eventually sending all
parameters to their combinatorial point values (2.5).

###  2.2. Boundary conditions

Figure 8. Quadrangular domain ${\mathcal{Q}}_{n}$ for the partition function
of the 20V model. We have indicated the horizontal spectral parameters
$z_{1},z_{2},...,z_{2n-1}$, the vertical ones $w_{1},w_{2},...,w_{n}$ and the
diagonal ones $t_{1},t_{2},...,t_{2n-1}$. We impose the following boundary
conditions: arrows pointing towards the domain on the E boundary, out of the
domain on the N and SW boundaries, and horizontal arrows point inward and
diagonal arrows point outward along the W boundary.

We now consider the configurations of the 20V model on the quadrangular domain
${\mathcal{Q}}_{n}={\mathcal{P}}_{n,n-1}$ depicted in Fig. 8, with a
horizontal north boundary (N), vertical west and east (W,E) boundaries and
diagonal south-west (SW) boundaries. We assign spectral parameters
${\mathbf{z}}=(z_{1},z_{2},...z_{2n-1})$ to horizontal lines (top to bottom),
${\mathbf{w}}=(w_{1},w_{2},...,w_{n})$ to vertical lines (left to right) and
${\mathbf{t}}=(t_{1},t_{2},...,t_{2n-1})$ to diagonal lines (bottom to top).
The corresponding fully inhomogeneous partition function is denoted by
$Z_{n}^{20V}[{\mathbf{z}};{\mathbf{t}};{\mathbf{w}}]$: it is the sum over all
configurations of the 20V model compatible with the imposed boundary
conditions, of the product of corresponding local weights
$\omega_{m}(z_{i},t_{j},w_{k})$ at the intersection of the three horizontal,
diagonal and vertical lines carrying spectral parameters $z_{i},t_{j},w_{k}$
respectively.

We also consider a semi-homogeneous version of this partition function
$Z_{n}^{20V}[z,t;{\mathbf{w}}]$ obtained by setting $z_{i}=z$ and $t_{i}=t$
for all $i$, while keeping the $w_{i}$ arbitrary.

The total number $Z_{n}^{20V}$ of configurations is obtained by considering
the abovementioned combinatorial point, namely further restricting $w_{i}=w$
for all $i$, and using the values (2.5).

###  2.3. Transformation to a mixed 6V model

Figure 9. Transformation of the original 20V model partition function
$Z_{n}^{20V}[z,t;{\mathbf{w}}]$ (a) into the alternating boundary mixed 6V
model partition function $Z_{n}^{6V}[z,t;{\mathbf{w}}]$ (d) (represented here
for $n=6$), by moving the diagonal lines, using the Yang-Baxter equation. The
marking of vertices corresponds to the weights of the three 6V models on the
three sublattices 1 (empty square), 2 (empty circle) and 3 (filled circle). In
(a), spectral parameters have uniform values $z$ (horizontal), $t$ (diagonal),
and arbitrary values $w_{i}$ (vertical). As a consequence, in (d) the
horizontal spectral parameters alternate between $z,t$ and the vertical ones
are $w_{1},w_{2},...,w_{n}$.

The semi-homogeneous partition function $Z_{n}^{20V}[z,t;{\mathbf{w}}]$ can be
identified with that of a special mixed 6V model on a rectangular grid, by use
of the Yang-Baxter equation, which allows to displace spectral lines while
keeping track of their intersection pattern (see Fig. 9 (a-d)).

Starting from (a), we first move up the $n$ uppermost diagonal lines, and down
the $n-1$ lowermost horizontal lines. Keeping track of their intersections
with the rest of the grid lines, these give rise to four triangular domains
with fixed arrow configurations (all with identical arrow orientations along
lines), due to the propagation of the boundary conditions via the ice rule.
The two top domains have $n(n+1)/2$ vertices each, while the two bottom ones
have $n(n-1)/2$.

Erasing them all, we get a new partition function. The original one is the
product of the trivial weights of all the erased vertices, namely:
$a_{2}(z,t)^{n^{2}}\prod_{i=1}^{n}b_{1}(z,w_{i})^{i-1}a_{3}(t,w_{i})^{i}$
times the new partition function with all the trivial vertices erased. (Here
$a_{i},b_{i},c_{i}$ denote the uniform weights of the three sublattices
$i=1,2,3$ 6V models (2.3)). The latter partition function is further
transformed into (c) by straightening the remaining diagonal lines, and
keeping track of their intersections. This gives rise to a last triangular
domain (on the right), again with fixed trivial arrow configurations. Erasing
the corresponding $n(n-1)/2$ trivial vertices gives rise to the partition
function of (d), which we denote by $Z_{n}^{m6V}[z,t;{\mathbf{w}}]$, up to an
extra factor of $a_{2}(z,t)^{n(n-1)/2}$. The acronym $m6V$ stands for mixed 6V
model partition function: it is defined on a rectangular grid of size
$(2n-1)\times n$ with 6V weights of sublattices 1 and 3 alternating on
horizontal lines, respectively with uniform horizontal spectral parameters $z$
and $t$, and vertical spectral parameters $w_{1},w_{2},...,w_{n}$, and with
arrows pointing alternatively in and out along the W border.

Recording all the weights of the erased vertices, we get the relation:

(2.6)
$Z_{n}^{20V}[z,t;{\mathbf{w}}]=\left(a_{2}(z,t)^{n(3n-1)/2}\,\prod_{i=1}^{n}b_{1}(z,w_{i})^{i-1}\,a_{3}(t,w_{i})^{i}\right)\,Z_{n}^{m6V}[z,t;{\mathbf{w}}]$

###  2.4. Transformation to a U-turn boundary 6V model

Figure 10. The U-turn boundary 6V model inhomogeneous partition function (a)
reduces when all $z_{i}=z=p^{-2}q^{-2}$ to the configurations of (b) where all
the arrow orientations at the U-turns point down, and where the top vertices
are all frozen in a b-type configuration. Erasing the latter and cutting the
U-turns leads to the partition function (c), in which horizontal spectral
parameters alternate between $z$ and $z^{-1}$ from bottom to top, while the
vertical ones are $w_{n},...,w_{2},w_{1}$ from left to right.

The U-turn boundary 6V model was independently studied by Tsuchiya [Tsu98] and
Kuperberg [Kup02]. The latter uses the 6V weights
$(A,B,C)(qz,q^{-1}w)/\sqrt{zw}$ (and up to a renaming of variables
$x=\sqrt{z/w}$, $a=q$, $b=p$). Here we drop the overall factor of
$1/\sqrt{zw}$ in the definition of the weights, which amounts to using the 6V
weights (2.3) of the sublattices 2 or 3.

Let $Z_{n}^{6V-U}[z_{1},z_{2},...,z_{n};w_{1},w_{2},...,w_{n};p]\equiv
Z_{n}^{6V-U}[{\mathbf{z}};{\mathbf{w}};p]$ denote the partition function of
the inhomogeneous 6V model on a rectangular grid of size $2n\times n$ with
alternating horizontal spectral parameters
$z_{1},z_{1}^{-1},...,z_{n},z_{n}^{-1}$ (from bottom to top), vertical
spectral parameters $w_{n},w_{n-1},...,w_{1}$ (from left to right), with
entering arrows along the W boundary, exiting arrows along the N and S
boundaries, and U-turn boundaries on the E, as described in Fig. 10 (a). The
marked dot on the U-turn indicates that the spectral parameter changes from
$z_{i}$ (bottom) to $z_{i}^{-1}$ (top) and that the configuration of the
corresponding U-turn receives a weight $Y(z_{i})$, where

(2.7)
$Y(z)=\left\\{\begin{matrix}pq^{-1}-p^{-1}qz&\raisebox{-8.5359pt}{\hbox{\epsfbox{Uboundary3.eps}}}\\\
\ &\ \\\
pqz-p^{-1}q^{-1}&\raisebox{-8.5359pt}{\hbox{\epsfbox{Uboundary4.eps}}}\end{matrix}\right.$

These weights obey Sklyanin’s Reflection Yang-Baxter equation [Skl88]. We now
relate the partition function of this model to that of the mixed 6V model of
previous section.

Setting all $z_{i}=z=p^{-2}q^{-2}$ forces each U-turn to have an arrow
pointing down, as indicated in Fig. 10 (b). However this imposes no constraint
on the vertical spectral parameters $w_{i}$, which we keep arbitrary. In the
resulting semi-homogeneous partition function, we may then cut out all the
marked dots on the U-turns, and moreover use the ice rule to dispose of the
top-most horizontal line, which gives rise to $n$ vertices with trivial b-type
configurations. Erasing these while keeping track of their weight leaves us
with the semi-homogeneous partition function $Z_{n}^{6V}[z;{\mathbf{w}}]$
corresponding to Fig. 10 (c), with alternating uniform horizontal spectral
parameters $z,z^{-1}$, and arbitrary vertical spectral parameters
${\mathbf{w}}=(w_{1},w_{2},...,w_{n})$:

(2.8)
$Z_{n}^{6V}[z;{\mathbf{w}}]:=\frac{Z_{n}^{6V-U}[z;{\mathbf{w}};p]}{(p^{2}-p^{-2})^{n}\prod_{i=1}^{n}(p-p^{-1}w_{i})}\
,\quad p=z^{-1/2}q^{-1},$

where we have divided by the top row b-type weights
$q^{-1}z^{-1}-qw_{i}=qp(p-p^{-1}w_{i})$ and by the U-turn weights
$pq^{-1}-p^{-1}qz=p^{-1}q^{-1}(p^{2}-p^{-2})$, using the relation
$z=p^{-2}q^{-2}$.

The partition function $Z_{n}^{6V}[z;{\mathbf{w}}]$ (2.8) may now be
identified with $Z_{n}^{m6V}[z,t;{\mathbf{w}}]$ of the previous section as
follows. Note first that a rotation of $180^{\circ}$ maps the configurations
of Fig. 10 (c) onto those of Fig. 9 (d) for general size $n$, and then that
this rotation flips all the local arrow orientations of the vertices, which
therefore keep the same weights (2.2). However a discrepancy subsists between
the weights in $Z_{n}^{m6V}[z,t;{\mathbf{w}}]$ and those in
$Z_{n}^{6V}[z;{\mathbf{w}}]$. Indeed, the 6V weights for
$Z_{n}^{m6V}[z,t;{\mathbf{w}}]$ alternate between those of the sublattice 1 on
odd rows and 3 on even rows. The weights on the even rows are those of the
sublattice 3 in both models, and we must therefore identify their spectral
parameters by setting $t=z^{-1}$. On odd rows, using the relation
$(a_{1},b_{1},c_{1})(z,w_{i})=(a_{3},b_{3},c_{3})(q^{-1}z,qw_{i})=q(a_{3},b_{3},c_{3})(q^{-2}z,w_{i})$,
we may identify these weights with $(a_{3},b_{3},c_{3})(z,w_{i})$ up to the
overall factor $q$, provided we redefine odd row parameters $z\to q^{2}z$
(while keeping $t=z^{-1}$ on even rows). This results in the following
relation between the semi-homogeneous partition functions:

$Z_{n}^{m6V}[q^{2}z,z^{-1};{\mathbf{w}}]=q^{n^{2}}Z_{n}^{6V}[z;{\mathbf{w}}]$

with $z=p^{-2}q^{-2}$. Together with (2.6), this leads to the relation:

(2.9)
$Z_{n}^{20V}[q^{2}z,z^{-1};{\mathbf{w}}]=\frac{q^{n^{2}}a_{2}(q^{2}z,z^{-1})^{n(3n-1)/2}\,\prod_{i=1}^{n}b_{1}(q^{2}z,w_{i})^{i-1}\,a_{3}(z^{-1},w_{i})^{i}}{(p^{2}-p^{-2})^{n}\prod_{i=1}^{n}(p-p^{-1}w_{i})}\,Z_{n}^{6V-U}[z;{\mathbf{w}};p]$

with $p=z^{-1/2}q^{-1}$.

To reach the combinatorial point (2.5), let us pick uniform 20V spectral
parameters so that $(q^{2}z,z^{-1},w_{i})=\delta(q^{6},1,q^{-6})$, namely
$z=q^{2},\delta=q^{-2},p=q^{-2}$ and $w_{i}=q^{-8}$ for all $i$, and finally
set $q=e^{i\pi/8}$. Using the homogeneity of the weights, this allows to
relate:

(2.10) $\displaystyle Z_{n}^{20V}$ $\displaystyle=$
$\displaystyle\left(\frac{{\alpha}}{\delta}\right)^{\frac{3n(3n-1)}{2}}\,Z_{n}^{20V}[q^{2}z,z^{-1},w]\big{|}_{z=q^{2};w=q^{-8}}$
$\displaystyle=$
$\displaystyle\frac{b_{1}(q^{4},q^{-8})^{\frac{n(n-1)}{2}}a_{2}(q^{4},q^{-2})^{\frac{n(3n-1)}{2}}a_{3}(q^{-2},q^{-8})^{\frac{n(n+1)}{2}}q^{n^{2}}}{\left(2^{5/6}q^{2}\right)^{\frac{3n(3n-1)}{2}}(q^{-4}-q^{4})^{n}(q^{-2}-q^{-6})^{n}}\,Z_{n}^{6V-U}[q^{2},q^{-8};q^{-2}]$
$\displaystyle=$ $\displaystyle
2^{-\frac{n(5n+3)}{4}}\,q^{-5n}\,Z_{n}^{6V-U}[q^{2},-1;q^{-2}]$

with ${\alpha}$ as in (2.5), and where the latter partition function
$Z_{n}^{6V-U}[q^{2},-1;q^{-2}]$ corresponds to taking uniform values
$z_{i}=q^{2}$ and $w_{i}=w=q^{-8}=-1$ for all $i$, and $p=q^{-2}$ in the
U-turn boundary partition function
$Z_{n}^{6V-U}[{\mathbf{z}},{\mathbf{w}};p]$.

######  Remark 2.2.

We note that neither $Z_{n}^{6V-U}[q^{2},-1;q^{-2}]$ nor
$Z_{n}^{6V}[q^{2},-1]$ are “combinatorial” in the sense that they count
objects. Indeed, they are weighted countings of configurations of the
corresponding models, including irrational weights (such as $\sqrt{2}$). For
instance, we may choose to normalize the weights of the $6V$ model by an
overall factor $\rho_{o}=q\sqrt{2}$ on odd rows such that
$\rho_{o}(A,B,C)(qz,q^{-1}w)=(1,\sqrt{2},1)$ and $\rho_{e}=q^{-1}\sqrt{2}$ on
even rows, such that $\rho_{e}(A,B,C)(qz^{-1},q^{-1}w)=(\sqrt{2},1,1)$ for
$z=q^{2}$ and $w=-1$. In that case, the $6V$ partition function
${\bar{Z}}_{n}^{6V}$ with these new weights reads
${\bar{Z}}_{n}^{6V}=\rho_{o}^{n^{2}}\,\rho_{e}^{n(n-1)}\,Z_{n}^{6V}[q^{2},-1]$.
Yet, using the above relations at the combinatorial point, we find that
${\bar{Z}}_{n}^{6V}=2^{n(n-1)/4}\,Z_{n}^{20V}$. So we should think of the $6V$
partition functions at hand as combinatorial tools rather than counting
combinatorial objects.

###  2.5. Inhomogeneous determinant formula

The result for the partition function of the U-turn boundary 6V model [Tsu98,
Kup02] reads as follows for our choice of weights. For short, we denote by
$(a,b,c)(z,w):=(a_{3},b_{3},c_{3})(z,w)=(A,B,C)(qz,q^{-1}w)$. Let $M_{U}(n)$
be the $n\times n$ matrix with entries

(2.11)
$[M_{U}(n)]_{i,j}=\frac{1}{a(z_{i},w_{j})b(z_{i},w_{j})}-\frac{1}{a(1,z_{i}w_{j})b(1,z_{i}w_{j})}$

Then the inhomogeneous partition function is given by:

$\displaystyle Z_{n}^{6V-U}[z_{1},z_{2},...,z_{n};w_{1},w_{2},...,w_{n};p]$
(2.12)
$\displaystyle\qquad=\det\big{(}M_{U}(n)\big{)}\times{\scriptstyle\frac{\left\\{\prod_{i=1}^{n}(p-p^{-1}w_{i})a(qz_{i},q^{-1}z_{i}^{-1})c(z_{i},w_{i})\right\\}\left\\{\prod_{i,j=1}^{n}a(z_{i},w_{j})b(z_{i},w_{j})a(1,z_{i}w_{j})b(1,z_{i}w_{j})\right\\}}{\left\\{\prod_{i=1}^{n}z_{i}^{n-1}\right\\}\left\\{\prod_{1\leq
i<j\leq n}(z_{i}-z_{j})(w_{j}-w_{i})\right\\}\left\\{\prod_{1\leq i\leq j\leq
n}(1-z_{i}z_{j})(1-w_{i}w_{j})\right\\}}}$

The aim of the following sections is to extract a homogeneous limit from this,
with relevant values of spectral parameters, to finally get a compact formula
for the partition function of the $20V$ model.

###  2.6. Homogeneous limit

Let us denote by $\Delta_{n}[z_{1},...,z_{n};w_{1},...,w_{n}]$ the quantity:

(2.13)
$\Delta_{n}[z_{1},...,z_{n};w_{1},...,w_{n}]:=\frac{\det\big{(}M_{U}(n)\big{)}}{\left\\{\prod_{1\leq
i<j\leq n}(z_{i}-z_{j})(w_{j}-w_{i})\right\\}\left\\{\prod_{1\leq i\leq j\leq
n}(1-z_{i}z_{j})(1-w_{i}w_{j})\right\\}}$

In view of the determinant formula (2.5), to deduce the homogeneous limit
$Z_{2n\times n}^{6V-U}[z,w;p]$ (2.8), we must now compute the limit

$\Delta_{n}[z,w]:=\lim_{z_{i}\to z\atop w_{i}\to
w}\Delta_{n}[z_{1},...,z_{n};w_{1},...,w_{n}]$

In fact, we are only interested in the special combinatorial point values
$z=q^{2},w=q^{8}=-1$ of the homogeneous parameters. In this section, we start
by computing $\Delta_{n}[z,-1]$.

Notation: Throughout the paper, for any function $f(u,v)$ with a power series
expansion around $(0,0)$, we denote by $f(u,v)|_{u^{i}v^{j}}$ the coefficient
of $u^{i}v^{j}$ in the series expansion of $f$, in other words
$f(u,v)|_{u^{i}v^{j}}:=\frac{\partial_{u}^{i}}{i!}\frac{\partial_{v}^{j}}{j!}f(0,0)$.

######  Theorem 2.3.

We have:

(2.14)
$\Delta_{n}[z,-1]=\frac{(-1)^{n(n-1)/2}}{2^{n}(1-z^{2})^{n(n+1)/2}}\det_{0\leq
i,j\leq n-1}\left(f_{U}(z;u,v)\Big{|}_{u^{i}v^{2j+1}}\right),$

where $f_{U}(z;u,v)$ denotes the function:

(2.15) $\displaystyle f_{U}(z;u,v)$ $\displaystyle:=$
$\displaystyle\frac{1}{a(z+u,v-1)b(z+u,v-1)}-\frac{1}{a(1,(z+u)(v-1))b(1,(z+u)(v-1))}$
$\displaystyle=$
$\displaystyle{\scriptstyle\frac{1}{(q^{-2}-q^{2})(z+u)}\left\\{\frac{q^{-1}}{a(z+u,v-1)}-\frac{q}{b(z+u,v-1)}-\frac{q^{-1}}{a(\frac{1}{z+u},v-1)}+\frac{q}{b(\frac{1}{z+u},v-1)}\right\\}}$

###### Proof.

The theorem is proved in two steps. We first keep $z_{i}$ generic, and compute
for $w=-1$:

$\Delta_{n}[z_{1},...,z_{n};-1]:=\lim_{w_{i}\to-1\atop
i=1,2,...,n}\frac{\det\big{(}M_{U}(n)\big{)}}{\left\\{\prod_{1\leq i<j\leq
n}(w_{j}-w_{i})\right\\}\left\\{\prod_{1\leq i\leq j\leq
n}(1-w_{i}w_{j})\right\\}}$

Setting $f[z;v]:=f_{U}(z;0,v)$, and setting $w_{i}=-1+v_{i}$, we may rewrite:

$\Delta_{n}[z_{1},...,z_{n};-1]:=\lim_{v_{i}\to 0\atop
i=1,2,...,n}\frac{\det_{1\leq i,j\leq
n}\big{(}f[z_{i};v_{j}]\big{)}}{\left\\{\prod_{1\leq i<j\leq
n}(v_{j}-v_{i})\right\\}\left\\{\prod_{1\leq i\leq j\leq
n}(v_{i}+v_{j})\right\\}}$

where in the last factors $1-w_{i}w_{j}=v_{i}+v_{j}-v_{i}v_{j}$ we have
dropped the higher order term $v_{i}v_{j}$ as the $v$’s tend to $0$.

For each $i$, we may expand $f[z_{i};v_{j}]$ in power series of $v_{j}$:
$f[z_{i};v_{j}]=\sum_{m\geq 0}f_{m}[z_{i}]v_{j}^{m}$, where the coefficients
read:

(2.16)
$f_{m}[z_{i}]={\scriptstyle\frac{1}{(q^{-2}-q^{2})z_{i}}\left\\{\left(\frac{1}{q^{2}z_{i}+1}\right)^{m+1}+\left(\frac{q^{2}z_{i}}{q^{2}z_{i}+1}\right)^{m+1}-\left(\frac{1}{q^{-2}z_{i}+1}\right)^{m+1}-\left(\frac{q^{-2}z_{i}}{q^{-2}z_{i}+1}\right)^{m+1}\right\\}}$

Let ${\mathbf{F}}[v]$ (resp. ${\mathbf{F}}_{m}$) denote the column vector with
entries $f[z_{i};v]$ (resp. $f_{m}[z_{i}]$) $i=1,2,...,n$. We have
${\mathbf{F}}[v]=\sum_{m\geq 0}v^{m}\,{\mathbf{F}}_{m}$. Moreover, the columns
${\mathbf{F}}_{m}$ are not linearly independent. More precisely we have the
following.

######  Lemma 2.4.

Let $c_{n}$, $n\in{\mathbb{Z}}_{+}$ be the coefficients in the series
expansion:

(2.17) $\frac{2}{1+e^{-x}}=\sum_{n\geq 0}c_{n}\,\frac{x^{n}}{n!}$

namely $c_{0}=1$, $c_{1}=\frac{1}{2}$, $c_{2}=0$, $c_{3}=-\frac{1}{4}$,
$c_{4}=0$, $c_{5}=\frac{1}{2}$, $c_{6}=0$, $c_{7}=\frac{17}{8}$, … Then for
all $n\geq 0$ we have:

(2.18) ${\mathbf{F}}_{2n}=\sum_{i=0}^{n-1}c_{2i+1}{2n+1\choose
2i+1}{\mathbf{F}}_{2n-2i-1}$

###### Proof.

For any function $f({\alpha})$ (say polynomial), introduce the shift operator
$S:f\mapsto Sf$, such that $(Sf)({\alpha})=f({\alpha}-1)$. Using the Taylor
expansion around ${\alpha}=0$, we may express $S$ as
$\exp(-\frac{d}{d{\alpha}})$. In particular, the identity
$(1+\exp(-\frac{d}{d{\alpha}}))f({\alpha})=f({\alpha})+f({\alpha}-1)$ is
easily inverted using the series expansion (2.17) in which $x$ is substituted
with the differential operator $\frac{d}{d{\alpha}}$:

$f({\alpha})=\frac{1}{2}\sum_{n\geq
0}c_{n}\frac{1}{n!}\frac{d^{n}}{d{\alpha}^{n}}\big{(}f({\alpha})+f({\alpha}-1)\big{)}$

Let us apply this to the polynomial $f({\alpha})={\alpha}^{2n+1}$ for $n\geq
0$. We get:

${\alpha}^{2n+1}=\frac{1}{2}\left\\{{\alpha}^{2n+1}+({\alpha}-1)^{2n+1}+\sum_{i=0}^{n}c_{2i+1}{2n+1\choose
2i+1}({\alpha}^{2n-2i}+({\alpha}-1)^{2n-2i})\right\\}$

which implies:

${\alpha}^{2n+1}+(1-{\alpha})^{2n+1}=\sum_{i=0}^{n}c_{2i+1}{2n+1\choose
2i+1}({\alpha}^{2n-2i}+(1-{\alpha})^{2n-2i})\ .$

Up to the overall prefactor $1/((q^{-2}-q^{2})z_{i})$, the $i$-th row of the
desired identity (2.18) follows immediately from this, as it takes precisely
the form:

$\displaystyle{\alpha}^{2n+1}+(1-{\alpha})^{2n+1}-(\beta^{2n+1}+(1-\beta)^{2n+1})$
$\displaystyle\quad=\sum_{i=0}^{n-1}c_{2i+1}{2n+1\choose
2i+1}(({\alpha}^{2n-2i}+({\alpha}-1)^{2n-2i})-(\beta^{2n-2i}+(\beta-1)^{2n-2i}))$

where ${\alpha}=\frac{1}{q^{2}z_{i}+1}$ and $\beta=\frac{1}{q^{-2}z_{i}+1}$. ∎

Note that as ${\mathbf{F}}_{0}=0$ we have the series expansion
${\mathbf{F}}[v]=\sum_{n\geq 1}{\mathbf{F}}_{n}v^{n}=:v{\mathbf{G}}[v]$. The
matrix with entries $f[z_{i};v_{j}]$ is made of successive columns
$v_{1}{\mathbf{G}}[v_{1}],...,v_{n}{\mathbf{G}}[v_{n}]$, therefore the
determinant of this matrix is $v_{1}v_{2}\cdots
v_{n}\det({\mathbf{G}}[v_{1}]{\mathbf{G}}[v_{2}]\cdots{\mathbf{G}}[v_{n}])$.
Consequently we rewrite:

$\frac{\det_{1\leq i,j\leq
n}\big{(}f[z_{i};v_{j}]\big{)}}{\left\\{\prod_{1\leq i<j\leq
n}(v_{j}-v_{i})\right\\}\left\\{\prod_{1\leq i\leq j\leq
n}(v_{i}+v_{j})\right\\}}=\frac{1}{2^{n}}\frac{\det({\mathbf{G}}[v_{1}]{\mathbf{G}}[v_{2}]\cdots{\mathbf{G}}[v_{n}])}{\prod_{1\leq
i<j\leq n}(v_{j}^{2}-v_{i}^{2})}=:D_{n}[{\mathbf{z}};{\mathbf{v}}]$

Let us now successively take $v_{1}\to 0$, $v_{2}\to 0$, … $v_{n}\to 0$. The
first limit $v_{1}\to 0$ amounts to taking ${\mathbf{G}}[0]={\mathbf{F}}_{1}$
as first column in the determinant and set $v_{1}=0$ in the denominator. To
perform the second, we must expand
${\mathbf{G}}[v_{2}]={\mathbf{F}}_{1}+{\mathbf{F}}_{2}v_{2}+{\mathbf{F}}_{3}v_{2}^{2}+O(v_{2}^{3})$
up to order $2$, as both constant and linear terms are columns proportional to
${\mathbf{F}}_{1}$ by Lemma 2.4. When we reach $v_{k}\to 0$, we must expand
${\mathbf{G}}[v_{k}]$ up to order $2k$, as all previous terms are linear
combinations of the previous columns in the determinant, as a consequence of
Lemma 2.4. This leads successively to:

$\displaystyle\lim_{v_{1}\to 0}D_{n}[{\mathbf{z}};{\mathbf{v}}]$
$\displaystyle=$
$\displaystyle\frac{1}{2^{n}}\frac{\det({\mathbf{F}}_{1}{\mathbf{G}}[v_{2}]\cdots{\mathbf{G}}[v_{n}])}{\prod_{i=2}^{n}v_{i}^{2}\prod_{2\leq
i<j\leq n}(v_{j}^{2}-v_{i}^{2})}$ $\displaystyle\lim_{v_{1},v_{2}\to
0}D_{n}[{\mathbf{z}};{\mathbf{v}}]$ $\displaystyle=$
$\displaystyle\frac{1}{2^{n}}\frac{\det({\mathbf{F}}_{1}{\mathbf{F}}_{3}{\mathbf{G}}[v_{3}]\cdots{\mathbf{G}}[v_{n}])}{\prod_{i=3}^{n}v_{i}^{4}\prod_{3\leq
i<j\leq n}(v_{j}^{2}-v_{i}^{2})}$ $\displaystyle\quad\vdots\ $
$\displaystyle\lim_{v_{1},v_{2},...,v_{n}\to
0}D_{n}[{\mathbf{z}};{\mathbf{v}}]$ $\displaystyle=$
$\displaystyle\frac{1}{2^{n}}\det({\mathbf{F}}_{1}{\mathbf{F}}_{3}\cdots{\mathbf{F}}_{2n-1})=\Delta_{n}[z_{1},...,z_{n};-1]$

We are now left with the second step, namely sending all $z_{i}$’s to $z$:

$\displaystyle\Delta_{n}[z,-1]$ $\displaystyle=$ $\displaystyle\lim_{z_{i}\to
z\atop i=1,2,...,n}\frac{\Delta_{n}[z_{1},...,z_{n};-1]}{\prod_{1\leq i<j\leq
n}(z_{i}-z_{j})\prod_{1\leq i\leq j\leq n}(1-z_{i}z_{j})}$ $\displaystyle=$
$\displaystyle\lim_{u_{i}\to 0\atop
i=1,2,...,n}\frac{1}{2^{n}(1-z^{2})^{n(n+1)/2}}\frac{\det({\mathbf{F}}_{1}{\mathbf{F}}_{3}\cdots{\mathbf{F}}_{2n-1})}{\prod_{1\leq
i<j\leq n}(u_{i}-u_{j})}$

where we have set $z_{i}=z+u_{i}$, and the column vector ${\mathbf{F}}_{2j+1}$
has entries $f_{2j+1}[z+u_{i}]$, $i=1,2,...,n$. Let us represent the matrix
$({\mathbf{F}}_{1}{\mathbf{F}}_{3}\cdots{\mathbf{F}}_{2n-1})$ by its rows, in
the form
$({\mathbf{R}}[u_{1}]{\mathbf{R}}[u_{2}]\cdots{\mathbf{R}}[u_{n}])^{t}$. Here,
the column vector ${\mathbf{R}}[u_{i}]$ is the transpose of the $i$-th row
vector of the original matrix, and ${\mathbf{R}}[u]$ has entries
$f_{2j+1}[z+u]$, $j=0,1,...,n-1$. As before, we write the series expansion
${\mathbf{R}}[u]=\sum_{n\geq 0}{\mathbf{R}}_{n}u^{n}$, and perform the
successive limits $u_{1}\to 0$, $u_{2}\to 0$, … $u_{n}\to 0$ in the quantity:

$D_{n}[{\mathbf{u}}]:=\frac{1}{2^{n}(1-z^{2})^{n(n+1)/2}}\frac{\det({\mathbf{R}}[u_{1}]{\mathbf{R}}[u_{2}]\cdots{\mathbf{R}}[u_{n}])^{t}}{\prod_{1\leq
i<j\leq n}(u_{i}-u_{j})}$

The vectors ${\mathbf{R}}_{i}$ turn out to be linearly independent, so that
each vector ${\mathbf{R}}[u_{j}]$ only has to be expanded up to order $j-1$,
and we end up with the formulas:

$\displaystyle\lim_{u_{1}\to 0}D_{n}[{\mathbf{u}}]$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n-1}}{2^{n}(1-z^{2})^{n(n+1)/2}}\frac{\det({\mathbf{R}}_{0}{\mathbf{R}}[u_{2}]\cdots{\mathbf{R}}[u_{n}])^{t}}{\prod_{i=2}^{n}u_{i}\prod_{2\leq
i<j\leq n}(u_{i}-u_{j})}$ $\displaystyle\lim_{u_{1},u_{2}\to
0}D_{n}[{\mathbf{u}}]$ $\displaystyle=$
$\displaystyle\frac{(-1)^{n-1+n-2}}{2^{n}(1-z^{2})^{n(n+1)/2}}\frac{\det({\mathbf{R}}_{0}{\mathbf{R}}_{1}{\mathbf{R}}[u_{3}]\cdots{\mathbf{R}}[u_{n}])^{t}}{\prod_{i=3}^{n}u_{i}^{2}\prod_{3\leq
i<j\leq n}(u_{i}-u_{j})}$ $\displaystyle\quad\vdots$
$\displaystyle\lim_{u_{1},u_{2},...,u_{n}\to 0}D_{n}[{\mathbf{u}}]$
$\displaystyle=$
$\displaystyle\frac{(-1)^{n(n-1)/2}}{2^{n}(1-z^{2})^{n(n+1)/2}}\det({\mathbf{R}}_{0}{\mathbf{R}}_{1}\cdots{\mathbf{R}}_{n})^{t}=\Delta_{n}[z;-1]$

The identity (2.14) follows by identifying the entries $i+1,j+1$
($i,j=0,1,...,n-1$) of the matrix
$({\mathbf{R}}_{0}{\mathbf{R}}_{1}\cdots{\mathbf{R}}_{n})^{t}$ with the
coefficient of $u^{i}v^{2j+1}$ of the series expansion of $f_{U}(u,v)$ around
$(0,0)$. Indeed, $({\mathbf{F}}_{2j+1})_{i+1}$ corresponds to the coefficient
of $v^{2j+1}$ in the series expansion of ${\mathbf{F}}[v]$ around $v=0$, and
$({\mathbf{R}}_{i})_{j+1}$ to the coefficient of $u^{i}$ in the expansion of
$({\mathbf{F}}_{2j+1})_{i+1}$ around $u=0$, after setting $z_{i}=z+u$. This
completes the proof of the theorem. ∎

To further the computation of $\Delta_{n}[z;-1]$, we need the following.

######  Lemma 2.5.

Let
${\tilde{f}}_{U}(u,v):=\gamma(u)\,\delta(v)\,f_{U}\Big{(}au{\alpha}(u),bv\beta(v)\Big{)}$,
with $a,b\in{\mathbb{C}}^{*}$, and where
${\alpha}(x),\beta(x),\gamma(x),\delta(x)$ are power series that converge for
small enough $x$, and take the value $1$ at $x=0$. Then we have:

$\Delta_{n}[z,-1]=\frac{(-1)^{n(n-1)/2}}{2^{n}(1-z^{2})^{n(n+1)/2}a^{n(n-1)/2}\,b^{n^{2}}}\det_{0\leq
i,j\leq n-1}\left({\tilde{f}}_{U}(u,v)\Big{|}_{u^{i}v^{2j+1}}\right)$

###### Proof.

Repeating the first step of the proof of Theorem 2.3 with ${\tilde{f}}_{U}$
substituted for $f_{U}$, we see that when we send $v_{j}\to 0$, the column
${\mathbf{F}}_{2j+1}$ gets multiplied by $b^{2j+1}$, but otherwise the
substitution $v_{j}\to bv_{j}\beta(v_{j})$, as well as the overall factor
$\delta(v_{j})$ only affect higher order terms of the expansion in $v_{j}$,
hence the limiting process yields the same result as before, up to an overall
factor $b^{1+3+\cdots+(2n-1)}=b^{n^{2}}$. The same happens for the second
step, where each vector ${\mathbf{R}}_{i}$ gets multiplied by $a^{i}$ leading
to an overall factor $a^{n(n-1)/2}$, while the substitution $u_{i}\to
au_{i}{\alpha}(u_{i})$, as well as the overall factor $\gamma(u_{i})$ again
only affect higher order terms in the expansion in $u_{i}$. The Lemma follows.
∎

We now compute the quantity $\Delta_{n}[q^{2},-1]$, corresponding to
$z=q^{2}$.

######  Theorem 2.6.

We have

(2.19) $\displaystyle\Delta_{n}[q^{2};-1]$ $\displaystyle=$
$\displaystyle\frac{q^{6n(n+1)}}{2^{\frac{n(7n+5)}{4}}}\det_{0\leq i,j\leq
n-1}\left(g_{U}(u,v)\Big{|}_{u^{i}v^{j}}\right)$ (2.20) $\displaystyle
g_{U}(u,v)$ $\displaystyle=$
$\displaystyle\frac{(1+u^{2})(1+2u-u^{2})}{(1-u^{2}v)\Big{(}(1-u)^{2}-v(1+u)^{2}\Big{)}}$

###### Proof.

We apply Lemma 2.5 with $z=q^{2}$, $a=2q^{6}$, $b=2q^{-4}$,
${\alpha}(u)=\frac{1}{1-q^{4}u}$, $\beta(v)=\frac{1}{1+q^{-4}v}$,
$\gamma(u)=\frac{1+q^{4}u}{1-q^{4}u}$, $\delta(v)=\beta(v)^{2}$, with the
result:

$\displaystyle{\tilde{f}}_{U}(u,v)$ $\displaystyle=$
$\displaystyle\frac{1+q^{4}u}{(1-q^{4}u)(1+q^{-4}v)^{2}}f_{U}\left(\frac{2q^{6}u}{1-q^{4}u},\frac{2q^{-4}v}{1+q^{-4}v}\right)$
$\displaystyle=$ $\displaystyle
v\,\frac{q^{4}}{1-q^{4}}\,\frac{(1+u^{2})(1+2u-u^{2})}{(1-u^{2}v^{2})\Big{(}(1-u)^{2}-v^{2}(1+u)^{2}\Big{)}}$
$\displaystyle=$ $\displaystyle-\frac{q^{-2}}{\sqrt{2}}\,v\,g_{U}(u,v^{2})$

Noting that the coefficient of $v^{2j+1}$ in ${\tilde{f}}_{U}(u,v)$
corresponds to the coefficient of $v^{j}$ in $g_{U}(u,v)$, the theorem follows
from assembling all the factors and using the fact that $q=e^{i\pi/8}$. ∎

###  2.7. The 20V model partition function

We now complete the calculation of the partition function for the $20V$ model
on the quadrangle $\mathcal{Q}_{n}$.

######  Theorem 2.7.

The partition function of the 20V model with uniform weights on the quadrangle
${\mathcal{Q}}_{n}$ reads:

(2.21) $\displaystyle Z_{n}^{20V}$ $\displaystyle=$ $\displaystyle\det_{0\leq
i,j\leq n-1}\left(g_{20V}(u,v)\Big{|}_{u^{i}v^{j}}\right)$ (2.22)
$\displaystyle g_{20V}(u,v)$ $\displaystyle=$
$\displaystyle\frac{(1+u^{2})(1+2u-u^{2})}{(1-u^{2}v)\Big{(}(1-u)^{2}-v(1+u)^{2}\Big{)}}=\frac{(\frac{1+u}{1-u})^{2}}{1-v(\frac{1+u}{1-u})^{2}}-\frac{u^{2}}{1-vu^{2}}$

###### Proof.

We use the result of Theorem 2.6 to compute the homogeneous limit of the
determinant formula (2.5) for $Z^{6V-U}_{n}[{\mathbf{z}},{\mathbf{w}};p]$,
with $z_{i}\to q^{2}$, $w_{i}\to-1$ and $p=q^{-2}$. First we assemble all the
remaining factors in $Z^{6V-U}_{n}[{\mathbf{z}},{\mathbf{w}};p]$ to find that:

$\displaystyle Z^{6V-U}_{n}[q^{2},-1]$ $\displaystyle=$
$\displaystyle\frac{1}{z^{n(n-1)}}\left((q^{2}-q^{-2})\sqrt{zw}\right(p^{-2}-p^{2})(p-p^{-1}w))^{n}$
$\displaystyle\times$
$\displaystyle\left((qz-q^{-1}w)(q^{-1}z-qw)(q-q^{-1}zw)(q^{-1}-qzw)\right)^{n^{2}}\,\Delta_{n}[z,w]|_{z\to
q^{2},w\to-1}$ $\displaystyle=$ $\displaystyle
q^{-2n(n-1)}(q^{2}-q^{-2})^{n}(q^{4}-q^{-4})^{n}(q^{2}+q^{-2})^{n}(8q^{4})^{n^{2}}\Delta_{n}[q^{2};-1]$
$\displaystyle=$ $\displaystyle
q^{n(2n-1)}2^{n(3n+2)}(-1)^{n}\Delta_{n}[q^{2};-1]$

Substituting the result of Theorem 2.6 for $\Delta_{n}[q^{2};-1]$, we finally
get:

$Z^{6V-U}_{n}=q^{5n}2^{\frac{n(5n+3)}{4}}\det_{0\leq i,j\leq
n-1}\left(g_{U}(u,v)\Big{|}_{u^{i}v^{j}}\right)$

The theorem follows from combining this with (2.10), and defining
$g_{20V}(u,v):=g_{U}(u,v)$. ∎

######  Remark 2.8.

We may interpret this result as follows. Let $P$ be the infinite matrix with
entries $P_{i,j}$, $i,j\in{\mathbb{Z}}_{+}$ generated by the series expansion
$g_{20V}(u,v)=\sum_{i,j\geq 0}u^{i}v^{j}P_{i,j}$ of (2.22). Then
$Z^{20V}_{n}=\det(P_{n})$ where $P_{n}$ is the truncation of the matrix $P$ to
it $n$ first rows and columns.

##  3\. Domino tilings of the Aztec triangle

###  3.1. Path model

Figure 11. The Aztec triangle $\mathcal{T}_{n}$ and a typical domino tiling
configuration (a), together with its bijectively associated family of non-
intersecting Schröder paths. In (b) we show the starting points of the $n$
paths: $(0,2i)$, $i=0,1,...,n-1$ and their endpoints $(2j+1,2j+1)$,
$j=0,1,..,n-1$.

The Aztec triangle $\mathcal{T}_{n}$ is defined as the planar domain of Figs.
11 (a,b), where the length of the vertical West boundary is $2n$. We study and
count the tiling configurations of this domain by means of $2\times 1$
dominos. The latter are in bijection with configurations of $n$ non-
intersecting lattice paths from the West boundary (starting points $(0,2i)$,
$i=1,2,...,n-1$) to the staircase East boundary (endpoints $(2j+1,2j+1)$,
$j=0,1,...,n-1$). These paths are Schröder paths, with three types of allowed
steps on ${\mathbb{Z}}^{2}$: up $(1,1)$, down $(1,-1)$ and horizontal $(2,0)$.

Let $Z_{n}^{DT}$ denote the total number of domino tiling configurations of
the Aztec triangle $\mathcal{T}_{n}$. The latter is easily computed by direct
application of the Lindström-Gessel-Viennot determinant formula [Lin73, GV85].
Denoting by $M_{i,j}$ the number of configurations of a single path from the
starting point $(0,2i)$ to the endpoint $(2j+1,2j+1)$, we have:

######  Lemma 3.1.

The partition function for domino tilings of the Aztec triangle
$\mathcal{T}_{n}$ reads:

$Z_{n}^{DT}=\det_{0\leq i,j\leq n-1}\left(M_{i,j}\right)$

######  Remark 3.2.

Note that $M_{i,j}$ is independent of $n$, as the shape of the Aztec triangle
domain imposes no further constraint on the paths apart from their fixed
starting and endpoints. The Lemma 3.1 states that $Z_{n}^{DT}=\det(M_{n})$,
where the $n\times n$ matrix $M_{n}$ may be viewed as the finite truncation to
the first $n$ rows and columns of the infinite matrix $M$ with entries
$M_{i,j}$, $i,j\in{\mathbb{Z}}_{+}$.

###  3.2. Generating functions and determinant formula

The generating function $f_{M}(u,v):=\sum_{i,j=0}^{\infty}v^{i}u^{j}\,M_{i,j}$
is given by the following.

######  Theorem 3.3.

We have

$f_{M}(u,v)=\sum_{i,j=0}^{\infty}v^{i}u^{j}\,M_{i,j}=\frac{1+u}{1-v-4uv-u^{2}v+u^{2}v^{2}}$

###### Proof.

We start from the generating series:

(3.1) $\sigma(u,v)=\frac{1}{1-u-v-uv}$

We may interpret $\sigma(u,v)$ as the partition function for arbitrary finite
length Schröder paths from the origin, with weight $u$ per up step, $v$ per
down step, and $uv$ per horizontal step. The position of the endpoint of the
path reads $(U+D+2H,U-D)$ where $U,D,H$ denote respectively the total numbers
of up, down, horizontal steps in the path. Such a path receives a weight
$u^{U}v^{D}(uv)^{H}=u^{U+H}v^{D+H}$. The number $\Sigma_{i,j}$ of paths from
the origin with fixed endpoint $(i,j)$, with $i-j$ even, therefore reads:
$\Sigma_{i,j}=\sigma(u,v)|_{u^{\frac{i+j}{2}}v^{\frac{i-j}{2}}}$.
Equivalently:

(3.2) $\Sigma_{i,j}=\oint\frac{dx}{2i\pi x}\frac{dy}{2i\pi
y}\,\frac{1}{x^{\frac{i+j}{2}}y^{\frac{i-j}{2}}}\,\frac{1}{1-x-y-xy}$

where the contour integral picks the residues at $0$. By a trivial translation
of the starting point, we may now interpret $M_{i,j}$ as the total number of
paths from the origin to $(2j+1,2j+1)-(0,2i)=(2j+1,2j+1-2i)$, hence
$M_{i,j}=\Sigma_{2j+1,2j+1-2i}$, or:

$M_{i,j}=\oint\frac{dx}{2i\pi x}\frac{dy}{2i\pi
y}\,\frac{1}{x^{2j+1-i}y^{i}}\,\frac{1}{1-x-y-xy}$

Changing integration variables to $u=y/x,v=x$, we get

$M_{i,j}=\oint\frac{du}{2i\pi u}\frac{dv}{2i\pi
v}\,\frac{1}{v^{2j+1}u^{i}}\,\frac{1}{1-v-uv-uv^{2}}=\frac{1}{1-v-uv-
uv^{2}}\Big{|}_{u^{i}v^{2j+1}}$

Finally, we remark that for any series $f(v)$, and any $j\in{\mathbb{Z}}_{+}$,
we have:

(3.3)
$f(v)|_{v^{2j+1}}=\frac{1}{2\sqrt{v}}\left(f(\sqrt{v})-f(-\sqrt{v})\right)\Big{|}_{v^{j}}$

and the theorem follows from the identity

$\frac{1}{2\sqrt{v}}\left(\frac{1}{1-(1+u)\sqrt{v}-uv}-\frac{1}{1+(1+u)\sqrt{v}-uv}\right)=\frac{1+u}{1-v-4uv-u^{2}v+u^{2}v^{2}}$

∎

This allows to translate the result of Lemma 3.1 into the following
determinant formula.

######  Theorem 3.4.

The partition function for domino tilings of the Aztec triangle
$\mathcal{T}_{n}$ reads:

(3.4) $\displaystyle Z^{DT}_{n}$ $\displaystyle=$ $\displaystyle\det_{0\leq
i,j\leq n-1}\left(f_{DT}(u,v)\Big{|}_{u^{i}v^{j}}\right)$ (3.5) $\displaystyle
f_{DT}(u,v)$ $\displaystyle=$
$\displaystyle\frac{1+u}{1-v-4uv-u^{2}v+u^{2}v^{2}}$

##  4\. 20V-Domino Tiling correspondence

###  4.1. Determinant relations

We wish to compare the results of Theorems 2.7 and 3.4. Using Remarks 2.8 and
3.2, we see that both enumeration results are expressed as the determinant of
a finite truncation of an infinite matrix. We have the following [DFL18].

######  Lemma 4.1.

Let $L$ (resp. $U$) be arbitrary infinite lower (resp. upper) triangular
matrices, with diagonal entries $1$, and $A$ any infinite matrix. Then we have
the following properties:

  1. (1)

For all $n\geq 1$, the truncation $(LAU)_{n}$ of the infinite matrix $LAU$ to
its first $n$ rows and columns is equal to $L_{n}A_{n}U_{n}$, the product of
the corresponding truncations of $L,A,U$ respectively.

  2. (2)

$\det(A_{n})=\det\big{(}(LAU)_{n}\big{)}$

###### Proof.

We have for all $i,j\in[0,n-1]$: $((LAU)_{n})_{i,j}=\sum_{\ell,m\geq
0}L_{i,\ell}A_{\ell,m}U_{m,j}$. But $L_{i,\ell}=0$ for $\ell>i$ hence in
particular for all $\ell\geq n$, as $i\leq n-1$. Similarly, $U_{m,j}=0$ for
$m\geq n$, and therefore we may rewrite
$((LAU)_{n})_{i,j}=\sum_{\ell,m=0}^{n-1}L_{i,\ell}A_{\ell,m}U_{m,j}=((L_{n}A_{n}U_{n})_{i,j}$,
and (1) follows. The statement (2) follows from
$\det(L_{n}A_{n}U_{n})=\det(A_{n})$ as $\det(L_{n})=\det(U_{n})=1$. ∎

######  Lemma 4.2.

Let $A,B$ be two infinite matrices with entries generated by the series
$f_{A}(u,v)=\sum_{i,j\geq 0}u^{i}v^{j}A_{i,j}$ and $f_{B}(u,v)=\sum_{i,j\geq
0}u^{i}v^{j}B_{i,j}$. Then the product $AB$ has for generating series the
convolution of $f_{A}$ and $f_{B}$, namely:

$f_{AB}(u,v)=f_{A}*f_{B}(u,v)=\oint\frac{dt}{2i\pi
t}f_{A}(u,t^{-1})f_{B}(t,v)$

where the contour integral picks up the residue at $t=0$.

###### Proof.

By direct calculation. ∎

###  4.2. Proof of the equivalence theorem

######  Theorem 4.3.

The partition function of the 20V DWBC3 model on the quadrangle
${\mathcal{Q}}_{n}$ and that of the domino tilings of the Aztec triangle
$\mathcal{T}_{n}$ coincide for all $n\geq 1$.

###### Proof.

By Remarks 2.8 and 3.2, we have $Z_{n}^{20V}=\det(P_{n})$,
$Z_{n}^{DT}=\det(M_{n})$ in terms of truncations of infinite square matrices
$P,M$ respectively generated by $f_{P}=g_{20V}$ (2.22) and $f_{M}=f_{DT}$
(3.5). Let $L$ be the infinite matrix with generating series

(4.1) $f_{L}(u,v):=\frac{1+2u-u^{2}}{1-u(1+v+uv)}\ .$

Then we have the following properties: (1) $L$ is lower triangular, with
diagonal elements $1$ and (2) $LM=P$. The property (1) follows from noticing
that we have a series identity of the form $f_{L}(u,v)=h(u,uv)$ where

$h(u,v)=\frac{1+2u-u^{2}}{1-u-v-uv}\ ,$

which shows that non-zero coefficients of $u^{i}v^{j}$ always have $j\geq i$.
Moreover the terms with $i=j$ correspond to the series expansion of
$h(0,uv)=1/(1-uv)$ and are all equal to $1$. Property (2) follows from Lemma
4.2:

$\displaystyle f_{LM}(u,v)$ $\displaystyle=$
$\displaystyle\oint\frac{dt}{2i\pi}\,\frac{1+2u-u^{2}}{t(1-u)-u(1+u)}f_{M}(t,v)$
$\displaystyle=$
$\displaystyle\frac{1+2u-u^{2}}{1-u}f_{M}\left(u\frac{1+u}{1-u},v\right)=f_{P}(u,v)$

easily checked by substituting the values $f_{P}=g_{20V}$ (2.22) and
$f_{M}=f_{DT}$ (3.5). Finally, by Lemma 4.1 we have
$\det(P_{n})=\det((LM)_{n})=\det(L_{n}M_{n})=\det(M_{n})$, and the theorem
follows. ∎

##  5\. Refined partition functions of the 20V model

###  5.1. Setting and weights

In this section, we consider the particular inhomogeneous partition function
$Z_{n}^{20V}[w]$ for the 20V model on the quadrangle of Fig. 8, corresponding
to a choice of uniform spectral parameters $z_{i}=q^{2}z=q^{4}$,
$i=1,2,...,n$, $t_{i}=z^{-1}=q^{-2}$, $i=1,2,...,2n-1$ and $w_{i}=q^{8}=-1$,
$i=1,2,...,n-1$, while $w_{n}=w$ is kept arbitrary. The corresponding weights
(2.1) are all equal to $1$, except those of the last column, which read:

$\displaystyle{\bar{\omega}}_{0}$ $\displaystyle=$
$\displaystyle\frac{(1-w)(q^{2}-q^{-2}w)}{2\sqrt{2}}$
$\displaystyle{\bar{\omega}}_{1}$ $\displaystyle=$
$\displaystyle\frac{(1-w)(q^{-2}-q^{2}w)}{2\sqrt{2}}$
$\displaystyle{\bar{\omega}}_{2}={\bar{\omega}}_{4}$ $\displaystyle=$
$\displaystyle\frac{1-w}{2}\,\sqrt{-w}$ (5.1)
$\displaystyle{\bar{\omega}}_{3}={\bar{\omega}}_{5}={\bar{\omega}}_{6}$
$\displaystyle=$ $\displaystyle\left(\frac{1-w}{2}\right)^{2}$

The quantity $Z_{n}^{20V}[w]$ gives access to refined 20V model partition
functions as explained in the next section.

###  5.2. Refined partition functions

Figure 12. A typical configuration of osculating Schröder paths on the
quadrangle $\mathcal{Q}_{n}$. The vertex labeled $k\in[1,2n-1]$ corresponds to
the first visit of any path to the last column. All vertices strictly above it
are in the empty configuration (weight $\omega_{0}$), while those strictly
below have a single vertical path going through (weight $\omega_{1}$). In
general configurations, the vertex $k$ itself could be in either of the two
indicated states (cameos on the right, with weights $\omega_{2}$ or
$\omega_{4}$).

Let us now relate the partition function $Z_{n}^{20V}[w]$ to refined
configuration numbers of the 20V model on the quadrangle ${\mathcal{Q}}_{n}$.
The osculating Schröder path configurations of the 20V model on
$\mathcal{Q}_{n}$ may be classified according to the configuration of the
vertices in the last column (see Fig. 12 for an illustration). Let $k$ denote
the position (labeled $1,2,...,2n-1$ from the bottom) of the unique vertex of
the last column where the top vertical edge is empty and the bottom vertical
edge is occupied by a path. Alternatively, this $k$-th vertex corresponds to
the point of entry of the topmost path into the rightmost vertical line, below
which the path goes down vertically until its endpoint. Note that all vertices
strictly above $k$ are in the empty configuration (weights $\omega_{0}$), and
all those strictly below in the configuration with exactly one vertical line
passing through (weights $\omega_{1}$). The vertex $k$ itself may be in either
of two configurations, corresponding to whether the path enters the last
column horizontally ($-$, weight $\omega_{4}$) or diagonally ($\backslash$,
weight $\omega_{2}$). Let $Z_{n,k}^{20V-}$ and $Z_{n,k}^{20V\backslash}$
denote the corresponding refined partition functions, with the obvious sum
rule at $w=-1$:

$\sum_{k=1}^{2n-1}Z_{n,k}^{20V-}+Z_{n,k}^{20V\backslash}=Z_{n}^{20V}$

In the presence of the spectral parameter $w$ in the last column, this
becomes:

(5.2)
$\sum_{k=1}^{2n-1}\left({\bar{\omega}}_{4}\,Z_{n,k}^{20V-}+{\bar{\omega}}_{2}\,Z_{n,k}^{20V\backslash}\right){\bar{\omega}}_{0}^{2n-k-1}{\bar{\omega}}_{1}^{k-1}=Z_{n}^{20V}[w]$

with ${\bar{\omega}}_{i}$ as in (5.1). Let us introduce a parameter:

(5.3)
$\tau:=\frac{{\bar{\omega}}_{1}}{{\bar{\omega}}_{0}}=\frac{q^{-2}-q^{2}w}{q^{2}-q^{-2}w}$

Further noting that ${\bar{\omega}}_{4}={\bar{\omega}}_{2}$, we get the
following expression for the generating function $Z_{n}^{20V}(\tau)$ of
refined 20V partition functions:

(5.4)
$Z_{n}^{20V}(\tau):=\sum_{k=1}^{2n-1}Z_{n,k}^{20V}\,\tau^{k-1}=\frac{q^{2}-q^{-2}w}{{\bar{\omega}_{0}}^{2n-1}\sqrt{-2w}}\,Z_{n}^{20V}[w]$

where we used the notation
$Z_{n,k}^{20V}=Z_{n,k}^{20V-}+Z_{n,k}^{20V\backslash}$ for the refined 20V
partition function corresponding to the leftmost path entering the rightmost
vertical line at position $k$. It is customary to also introduce the
generating function $h_{n}^{20V}(\tau)$ for normalized “one-point functions”
$\frac{Z_{n,k}^{20V}}{Z_{n}^{20V}}$:

(5.5)
$h_{n}^{20V}(\tau):=\sum_{k=1}^{2n-1}\frac{Z_{n,k}^{20V}}{Z_{n}^{20V}}\,\tau^{k-1}=\frac{Z_{n}^{20V}(\tau)}{Z_{n}^{20V}}=\frac{q^{2}-q^{-2}w}{{\bar{\omega}_{0}}^{2n-1}\sqrt{-2w}}\,\frac{Z_{n}^{20V}[w]}{Z_{n}^{20V}}$

###  5.3. Transformations to the 6V model

Repeating the operations of Sections 2.3 and 2.4, and recording all the
weights of trivial vertices, we arrive at:

(5.6) $Z_{n}^{20V}[w]={\scriptstyle
q^{n^{2}}b_{1}(q^{4},w)^{n-1}a_{3}(q^{-2},w)^{n}a_{2}(q^{4},q^{-2})^{\frac{n(3n-1)}{2}}b_{1}(q^{4},-1)^{\frac{(n-1)(n-1)}{2}}\,a_{3}(q^{-2},-1)^{\frac{n(n-1)}{2}}}\,Z_{n}^{6V}[w]$

where $Z_{n}^{6V}[w]$ denotes the partition function function of the 6V model
with boundary conditions as in Fig. 10 (c), with the vertex weights
$(A,B,C)(qz_{i},q^{-1}w_{j})$ identical to those of the sublattices 2 and 3,
with vertical spectral parameters $w_{1}=\cdots=w_{n-1}=-1$, $w_{n}=w$, and
horizontal spectral parameters alternating between the value $z=q^{2}$ on odd
rows and $z^{-1}=q^{-2}$ on even rows. Eq. 5.6 boils down to:

(5.7)
$\frac{Z_{n}^{20V}[w]}{Z_{n}^{20V}}=\left(\frac{1-w}{2}\right)^{2n-1}\,\frac{Z_{n}^{6V}[w]}{Z_{n}^{6V}}$

trivially satisfied for $w=-1$. Next, (2.8) for $p=q^{-2}$ implies:

(5.8)
$Z_{n}^{6V}[w]=\frac{Z_{n}^{6V-U}[w]}{(p^{2}-p^{-2})^{n}(p+p^{-1})^{n-1}(p-p^{-1}w)}$

and equivalently:

(5.9)
$\frac{Z_{n}^{20V}[w]}{Z_{n}^{20V}}=\frac{\sqrt{2}}{q^{-2}-q^{2}w}\,\left(\frac{1-w}{2}\right)^{2n-1}\,\frac{Z_{n}^{6V-U}[w]}{Z_{n}^{6V-U}}$

###  5.4. Refined partition functions of the 6V model

Figure 13. A typical configuration of osculating paths for the $6V$ model on a
rectangular grid of size $(2n-1)\times n$. The boundary conditions match those
of Fig. 10 (c) upon global rotation by 180∘. The horizontal spectral
parameters alternates between the values $z=q^{2}$ on odd lines and
$z^{-1}=q^{-2}$ on even lines. The vertical parameters are $w_{i}=q^{8}=-1$
except in the last column, where $w_{n}=w$. The vertex labeled $k\in[1,2n-1]$
corresponds to the first visit of any path to the last column. All vertices
strictly above it are in the empty configuration (weight $a_{e},a_{o}$ on
even/odd rows), while those strictly below have a single vertical path going
through (weight $b_{e},b_{o}$ on even/odd rows). In general configurations,
the vertex $k$ itself could be in either of the two indicated states (cameos
on the right, with weights $c_{e}$ if $k$ is even or $c_{o}$ if $k$ is odd).

We now turn to the partition function $Z_{n}^{6V}[w]$ of the 6V model. To
easier compare it to that of the 6V model with U-turn boundaries, let us
rotate the picture of Fig. 10 (c) by 180∘. The rows are now labeled
$1,2,...,2n-1$ from bottom to top, and the non-trivial spectral parameter $w$
is attached to the rightmost vertical line. We consider the osculating path
configurations of the model (see Fig. 5 bottom), with $n$ paths starting on
odd rows along the left boundary and ending on the S boundary (see Fig. 13 for
an illustration). We note that the topmost path must first visit the rightmost
vertical line at a vertex corresponding to some row $k$, and then continue
along the vertical until its endpoint. The configurations along the rightmost
vertical line have therefore b-type weights at all positions strictly under
$k$, and a-type weights at all positions strictly above $k$, while the vertex
at position $k$ is of type c. When collecting the weights however, we must
distinguish the cases according to the parity of $k$, as odd and even
horizontal spectral parameters are different. Using the odd row weights:
$(A,B,C)(q^{3},q^{-1}w)$ and even row weights $(A,B,C)(q^{-1},q^{-1}w)$, we
get the following relative weights (ratios of the weights at generic $w$ to
those at $w=-1$) in the last column:

$\begin{matrix}{\bar{a}}_{o}=\frac{q^{2}-q^{-2}w}{\sqrt{2}}&{\bar{b}}_{o}=\frac{1-w}{2}&{\bar{c}}_{o}=\sqrt{-w}\\\
{\bar{a}}_{e}=\frac{1-w}{2}&{\bar{b}}_{e}=\frac{q^{-2}-q^{2}w}{\sqrt{2}}&{\bar{c}}_{e}=\sqrt{-w}\end{matrix}$

where the subscript $e/o$ stands for even/odd row.

Let $Z_{n,k}^{6V}$ denote the refined partition function444As already
mentioned in Remark 2.2, the quantities $Z_{n,k}^{6V}$ are not integers as the
weighting of the $6V$ model is not uniform and involves $\sqrt{2}$ weights.
Here and in the following we have dropped the arguments $[z=q^{2},w=-1]$
indicating that we are at the combinatorial point, and we use the notation
$[w]$ to indicate a non-trivial value $w$ for the $n$-th vertical spectral
parameter, while all other spectral parameters have the values of the
combinatorial point. corresponding to paths first visiting the rightmost
vertical at position $k$. Apart from the trivial sum rule at $w=-1$:

$\sum_{k=1}^{2n-1}Z_{n,k}^{6V}=Z_{n}^{6V}$

we have in the presence of a non-trivial $w$:

$\sum_{k=1}^{n}Z_{n,2k-1}^{6V}({\bar{b}}_{o}{\bar{b}}_{e})^{k-1}{\bar{c}}_{o}({\bar{a}}_{e}{\bar{a}}_{o})^{n-k}+\sum_{k=1}^{n-1}Z_{n,2k}^{6V}({\bar{b}}_{o}{\bar{b}}_{e})^{k-1}{\bar{b}}_{o}{\bar{c}}_{e}{\bar{a}}_{o}({\bar{a}}_{e}{\bar{a}}_{o})^{n-k-1}=Z_{n}^{6V}[w]$

Noting that ${\bar{c}}_{e}={\bar{c}}_{o}$,
${\bar{b}}_{o}{\bar{a}}_{o}={\bar{a}}_{e}{\bar{a}}_{o}={\bar{\omega}}_{0}$ and
${\bar{b}}_{o}{\bar{b}}_{e}={\bar{\omega}}_{1}$, and using the parameter
$\tau$ of (5.3), we finally get an expression for the generating function of
the sums of refined partition functions $Z_{n,2k-1}^{6V}+Z_{n,2k}^{6V}$:

(5.10)
$Z_{n}^{6V}(\tau):=\sum_{k=1}^{n}(Z_{n,2k-1}^{6V}+Z_{n,2k}^{6V})\,\tau^{k-1}=\frac{1}{{\bar{\omega}_{0}}^{n-1}\sqrt{-w}}\,Z_{n}^{6V}[w]$

or equivalently the generating function $h_{n}^{6V}(\tau)$ for sums of one-
point functions $\frac{Z_{n,2k-1}^{6V}+Z_{n,2k}^{6V}}{Z_{n}^{6V}}$:

(5.11)
$h_{n}^{6V}(\tau):=\sum_{k=1}^{n}\frac{Z_{n,2k-1}^{6V}+Z_{n,2k}^{6V}}{Z_{n}^{6V}}\,\tau^{k-1}=\frac{Z_{n}^{6V}(\tau)}{Z_{n}^{6V}}=\frac{1}{{\bar{\omega}_{0}}^{n-1}\sqrt{-w}}\,\frac{Z_{n}^{6V}[w]}{Z_{n}^{6V}}$

where we set $Z_{n,2n}^{6V}=0$ by convention.

Combining (5.5), (5.11) and the relation (5.7), we finally get the following
relation between one-point generating functions of the $20V$ and $6V$ models:

(5.12)
$h_{n}^{20V}(\tau)={\bar{\omega}}_{0}^{n-1}\,\frac{q^{2}-q^{-2}w}{\sqrt{2}}\left(\frac{1-w}{2\,{\bar{\omega}}_{0}}\right)^{2n-1}\,h_{n}^{6V}(\tau)=\left(\frac{1+\tau}{2}\right)^{n-1}\,h_{n}^{6V}(\tau)$

where we have identified the quantity
$\frac{1-w}{\sqrt{2}(q^{2}-q^{-2}w)}=\frac{1+\tau}{2}$.

###  5.5. Relation to the inhomogeneous U-turn boundary partition function

Let us finally relate the above quantities to the U-turn boundary partition
function $Z_{n}^{6V-U}[w]$. First we note that (5.8) implies:

$\frac{Z_{n}^{6V}[w]}{Z_{n}^{6V}}=\frac{\sqrt{2}}{q^{-2}-q^{2}w}\,\frac{Z_{n}^{6V-U}[w]}{Z_{n}^{6V-U}}$

and moreover that (2.5) implies the relation:

$\frac{Z_{n}^{6V-U}[w]}{Z_{n}^{6V-U}}=\sqrt{-w}\left(\frac{1-w}{2}\right)^{2n}\left(\frac{q^{-2}-q^{2}w}{\sqrt{2}}\right)^{n+1}\left(\frac{q^{2}-q^{-2}w}{\sqrt{2}}\right)^{n}\frac{\Delta_{n}[w]}{\Delta_{n}}$

where we denote for short $\Delta_{n}[w]$ the quantity
$\Delta_{n}[z_{1},...,z_{n};w_{1},...,w_{n}]$ (2.13) with all $z_{i}=q^{2}$
and all $w_{i}=-1$ except $w_{n}=w$, and $\Delta_{n}:=\Delta_{n}[-1]$.
Consequently, we have:

(5.13) $\displaystyle h_{n}^{6V}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{{\bar{\omega}_{0}}^{n-1}\sqrt{-w}}\frac{\sqrt{2}}{q^{-2}-q^{2}w}\,\frac{Z_{n}^{6V-U}[w]}{Z_{2n\times
n}^{6V-U}}$ $\displaystyle=$
$\displaystyle\left(\frac{1-w}{2}\right)^{n+1}\left(\frac{q^{2}-q^{-2}w}{\sqrt{2}}\right)\left(\frac{q^{-2}-q^{2}w}{\sqrt{2}}\right)^{n}\frac{\Delta_{n}[w]}{\Delta_{n}}$

Similarly:

(5.14)
$h_{n}^{20V}(\tau)=\left(\frac{1-w}{2}\right)^{2n}\left(\frac{q^{2}-q^{-2}w}{\sqrt{2}}\right)^{2}\left(\frac{q^{-2}-q^{2}w}{q^{2}-q^{-2}w}\right)^{n}\frac{\Delta_{n}[w]}{\Delta_{n}}$

###  5.6. Refined U-turn boundary

We now turn to the calculation of the refined determinant $\Delta_{n}[w]$. We
start with a refinement of Theorem 2.3. We consider the semi-homogeneous limit

$\Delta_{n}^{\prime}[z,w]:=\lim_{z_{1},...,z_{n}\to z\atop
w_{1},...,w_{n-1}\to-1,w_{n}\to w}\Delta_{n}[z_{1},...,z_{n};w_{1},...,w_{n}]$

As in the proof of Theorem 2.3, we proceed in two steps. First, we take the
successive limits $w_{1},w_{2},...,w_{n-1}\to-1$. This step is rigorously
identical, as only the limit $w_{n}\to w$ is different. Let us consider:

$\displaystyle\Delta_{n}^{\prime}[{\mathbf{z}},w]$ $\displaystyle:=$
$\displaystyle\lim_{w_{1},...,w_{n-1}\to-1\atop w_{n}\to
w}\Delta_{n}[z_{1},...,z_{n};w_{1},...,w_{n}]$ $\displaystyle=$
$\displaystyle\lim_{v_{1},v_{2},...,v_{n}\to
0}\frac{\det\left({\mathbf{F}}[v_{1}]{\mathbf{F}}[v_{2}]\cdots{\mathbf{F}}[v_{n-1}]{\tilde{\mathbf{F}}}[v_{n}]\right)}{2^{n-1}(w+1)^{2n-2}(1-w^{2})\prod_{1\leq
i<j\leq n-1}(v_{j}^{2}-v_{i}^{2})\prod_{i=1}^{n-1}v_{i}}$

where ${\mathbf{F}}[v]$ is as in Sect. 2.6, and where
${\tilde{\mathbf{F}}}[v]:={\mathbf{F}}[v+1+w]$, which accounts for taking the
limit $w_{n}\to w$ instead of $-1$ in the last column. Writing
${\mathbf{F}}[v]=v{\mathbf{G}}[v]$ as before, and performing successively the
limits $v_{1}\to 0,...,v_{n-1}\to 0$, we end up with

$\Delta_{n}^{\prime}[{\mathbf{z}},w]=\frac{\det\left({\mathbf{F}}_{1}{\mathbf{F}}_{2}\cdots{\mathbf{F}}_{n-1}{{\mathbf{F}}}[w+1]\right)}{2^{n-1}(w+1)^{2n-1}(1-w)}$

The second step consists in computing:

$\displaystyle\Delta_{n}^{\prime}[z,w]$ $\displaystyle=$
$\displaystyle\lim_{z_{1},...,z_{n}\to z}\Delta_{n}^{\prime}[{\mathbf{z}},w]$
$\displaystyle=$ $\displaystyle\lim_{u_{1},...,u_{n}\to
0}\frac{\det\left({\mathbf{F}}_{1}{\mathbf{F}}_{2}\cdots{\mathbf{F}}_{n-1}{{\mathbf{F}}}[w+1]\right)}{2^{n-1}(w+1)^{2n-1}(1-w)(1-z^{2})^{n(n+1)/2}\prod_{1\leq
i<j\leq n}(u_{i}-u_{j})}$

Like in Sect. 2.6, let us denote by
${\tilde{\mathbf{R}}}[u_{1}],...,{\tilde{\mathbf{R}}}[u_{n}]$ the transpose of
the rows of the matrix
$\left({\mathbf{F}}_{1}{\mathbf{F}}_{2}\cdots{\mathbf{F}}_{n-1}{{\mathbf{F}}}[w+1]\right)$,
where ${\tilde{\mathbf{R}}}[u]$ has entries
$(f_{1}[z+u],f_{3}[z+u],...,f_{2n-3}[z+u],f[z+u;w+1])$, with
$f[z+u;w+1]=f_{U}(u,w+1)$ (2.15) and $f_{i}$ as in (2.16). Expanding again
${\tilde{\mathbf{R}}}[u]=\sum_{m\geq 0}u^{m}\,{\tilde{\mathbf{R}}}_{m}$, and
performing the successive limits $u_{1}\to 0,...,u_{n}\to 0$, we arrive at:

$\Delta_{n}^{\prime}[z,w]=(-1)^{n(n-1)/2}\,\frac{\det\left(({\tilde{\mathbf{R}}}_{1}{\tilde{\mathbf{R}}}_{2}\cdots{\tilde{\mathbf{R}}}_{n})^{t}\right)}{2^{n-1}(w+1)^{2n-1}(1-w)(1-z^{2})^{n(n+1)/2}}$

The entries $({\tilde{\mathbf{R}}}_{m})_{i}$ of the vector
${\tilde{\mathbf{R}}}_{m}$ are easily identified as
$f_{U}(u,v)|_{u^{i}v^{2m-1}}$ for $m=1,2,...,n-1$ and $f_{U}(u,w+1)|_{u^{i}}$
for $m=n$. This result may be recast into the following.

######  Lemma 5.1.

We have

$\Delta_{n}^{\prime}[z,w]=(-1)^{n(n-1)/2}\,\frac{\det_{1\leq i,j\leq
n-1}\left(f_{U}(z,w;u,v)|_{u^{i}v^{2j+1}}\right)}{2^{n-1}(w+1)^{2n-1}(1-w)(1-z^{2})^{n(n+1)/2}}$

in terms of the function:

(5.15)
$f_{U}(z,w;u,v):=f_{U}(z;u,v)+v^{2n-1}\left(f_{U}(z;u,w+1)-f_{U}(z;u,v)|_{v^{2n-1}}\right)$

(Note the subtraction of the contribution of order $v^{2n-1}$ of
$f_{U}(z;u,v)$, to avoid over-counting.). We are now ready to compute the
quantity $\Delta_{n}[w]=\Delta_{n}^{\prime}[q^{2},w]$ corresponding to
$z=q^{2}$.

######  Theorem 5.2.

We have the following determinant formula:

(5.16) $\displaystyle\quad\qquad\Delta_{n}[w]$ $\displaystyle=$
$\displaystyle\frac{q^{6n(n+1)}}{2^{\frac{n(7n+5)}{4}}}\left(\frac{2}{1-w}\right)^{2n}\,\det_{0\leq
i,j\leq n-1}\left(g_{U}^{\prime}(u,v)|_{u^{i}v^{j}}\right)$ (5.17)
$\displaystyle\quad\qquad g_{U}^{\prime}(u,v)$ $\displaystyle=$ $\displaystyle
g_{U}(u,v)+v^{n-1}\left(\frac{1+u}{1-u}\right)^{2n}\left\\{\frac{4(1-u)^{2}}{(1-u)^{2}(1-w)^{2}+(1+u)^{2}(1+w)^{2}}-1\right\\}$

with $g_{U}(u,v)$ as in (2.20).

###### Proof.

Like in Sect. 2.6, we apply Lemma 2.5 to the function $f_{U}(q^{2},w;u,v)$
(5.15), with the same choices of $a,b,{\alpha},\beta,\gamma,\delta$. More
precisely we apply the substitutions:

(5.18) $u\to\frac{2q^{6}u}{1-q^{4}u},\qquad
v\to\frac{2q^{-4}v}{1+q^{-4}v},\qquad w=\frac{2q^{-4}x}{1+q^{-4}x}-1$

and we multiply it by an overall factor of
$\frac{1+q^{4}u}{(1-q^{4}u)(1+q^{-4}v)^{2}}$. The last substitution in (5.18)
is simply a change of variable $w\to x$, which allows to treat $v$ and $w+1$
similarly. We first compute the effect of this transformation on the term
$v^{2n-1}f_{U}(z;u,w+1)$, up to higher orders in $v$:

$\displaystyle\frac{1+q^{4}u}{(1-q^{4}u)(1+q^{-4}v)^{2}}(2q^{-4}v)^{2n-1}f_{U}\left(q^{2};\frac{2q^{6}u}{1-q^{4}u},\frac{2q^{-4}x}{1+q^{-4}x}\right)$
$\displaystyle\quad=(2v)^{2n-1}q^{4}(-1)^{n}\frac{q^{4}}{1-q^{4}}\,(1+q^{-4}x)^{2}\,x\,g_{U}(u,x^{2})+O(v^{2n})$

with $g_{U}(u,v)$ as in (2.20). The complete function $f_{U}(q^{2},w;u,v)$ is
therefore transformed into:

$\displaystyle{\tilde{f}}_{U}(q^{2},x;u,v)$ $\displaystyle=$ $\displaystyle
v\frac{q^{4}}{1-q^{4}}\left\\{g_{U}(u,v^{2})\right.$
$\displaystyle\qquad\left.+v^{2n-2}\left(2^{2n-1}q^{4}(-1)^{n}(1+q^{-4}x)^{2}x\,g_{U}(u,x^{2})-g_{U}(u,v^{2})|_{v^{2n-2}}\right)\right\\}$

In this last expression we may replace $xg_{U}(u,x^{2})=\sum_{m\geq
0}x^{2m+1}\left\\{\left(\frac{1+u}{1-u}\right)^{2m+2}-u^{2m+2}\right\\}$ with
the truncated expansion

$x{\bar{g}}(u,x^{2})=\sum_{m\geq
n-1}x^{2m+1}\left\\{\left(\frac{1+u}{1-u}\right)^{2m+2}-u^{2m+2}\right\\}=x^{2n-1}\left(\frac{1+u}{1-u}\right)^{2n}\,\frac{1}{1-x^{2}\left(\frac{1+u}{1-u}\right)^{2}}+O(u^{2n})$

Indeed, the first $n-1$ terms in the series produce columns that are linear
combinations of the first $n-1$ columns in the determinant, and can therefore
be omitted without affecting the result. Finally we restore the variable $w$
by inverting the change of variables in (5.18) into $x=q^{4}\frac{1+w}{1-w}$.
The corresponding function finally reads:

$\displaystyle{\tilde{f}}_{U}^{\prime}(q^{2},w;u,v)=-\frac{vq^{-2}}{\sqrt{2}}\left\\{g_{U}(u,v^{2})\right.$
$\displaystyle\quad\left.+v^{2n-2}\left(\frac{1+u}{1-u}\right)^{2n}\left(\left(\frac{2(1+w)}{1-w}\right)^{2n-1}\frac{4(1-u)^{2}}{(1-u)^{2}(1-w)^{2}+(1+u)^{2}(1+w)^{2}}-1\right)\right\\}$
$\displaystyle\quad=:-\frac{vq^{-2}}{\sqrt{2}}{\tilde{g}}_{U}^{\prime}(u,v^{2})$

By Lemma 2.5 we deduce:

$\displaystyle\Delta_{n}[w]$ $\displaystyle=$
$\displaystyle\frac{q^{6n(n+1)}}{2^{\frac{n(7n+5)}{4}}\frac{1-w}{2}(1+w)^{2n-1}}\,\det_{0\leq
i,j\leq n-1}\left({\tilde{g}}_{U}^{\prime}(u,v)|_{u^{i}v^{j}}\right)$
$\displaystyle=$
$\displaystyle\frac{q^{6n(n+1)}}{2^{\frac{n(7n+5)}{4}}}\left(\frac{2}{1-w}\right)^{2n}\,\det_{0\leq
i,j\leq n-1}\left({g}_{U}^{\prime}(u,v)|_{u^{i}v^{j}}\right)$

in which we have divided explicitly the last column by
$\left(\frac{2(1+w)}{1-w}\right)^{2n-1}$, thus allowing to replace
${\tilde{g}}_{U}^{\prime}(u,v)$ with ${g}_{U}^{\prime}(u,v)$. The theorem
follows. ∎

###  5.7. Refined partition functions for 20V and 6V models

Theorem 5.2 provides the following formula for the ratio
$\Delta_{n}[w]/\Delta_{n}$:

$\frac{\Delta_{n}[w]}{\Delta_{n}}=\Big{(}\frac{2}{1-w}\Big{)}^{2n}\,\,\frac{\det_{0\leq
i,j\leq n-1}\left(g_{U}^{\prime}(u,v)|_{u^{i}v^{j}}\right)}{\det_{0\leq
i,j\leq n-1}\left(g_{U}(u,v)|_{u^{i}v^{j}}\right)}$

in terms of the functions $g_{U}(u,v)$ (2.20) and $g_{U}^{\prime}(u,v)$
(5.17).

Using the relations (5.13) and (5.14) and the result of Theorem 5.2, we
immediately get the following expressions for the one-point function
generating functions of the 6V and 20V models, in terms of the variable
$\tau=\frac{1-q^{4}w}{q^{4}-w}$ (i.e. $w=\frac{1-q^{4}\tau}{q^{4}-\tau}$), and
of the function $g_{20V}=g_{U}$:

######  Theorem 5.3.

The one-point generating functions for the $6V$ and $20V$ models read for all
$n\geq 1$:

(5.19) $\displaystyle h_{n}^{6V}(\tau)$ $\displaystyle=$
$\displaystyle\frac{\det_{0\leq i,j\leq n-1}\left(g_{6V}^{\rm
ref}(u,v)|_{u^{i}v^{j}}\right)}{\det_{0\leq i,j\leq
n-1}\left(g_{20V}(u,v)|_{u^{i}v^{j}}\right)}$ (5.20) $\displaystyle
h_{n}^{20V}(\tau)$ $\displaystyle=$ $\displaystyle\frac{\det_{0\leq i,j\leq
n-1}\left(g_{20V}^{\rm ref}(u,v)|_{u^{i}v^{j}}\right)}{\det_{0\leq i,j\leq
n-1}\left(g_{20V}(u,v)|_{u^{i}v^{j}}\right)}$

in terms of the functions:

(5.21) $\displaystyle\qquad g_{6V}^{\rm ref}(u,v)$ $\displaystyle=$
$\displaystyle
g_{20V}(u,v)+v^{n-1}\left(\frac{1+u}{1-u}\right)^{2n}\left\\{\left(\frac{2}{1+\tau}\right)^{n-1}\frac{\tau^{n}\,(1-u)^{2}}{(\tau-u)(1-\tau
u)}-1\right\\}$ (5.22) $\displaystyle\qquad g_{20V}^{\rm ref}(u,v)$
$\displaystyle=$ $\displaystyle
g_{20V}(u,v)+v^{n-1}\left(\frac{1+u}{1-u}\right)^{2n}\left\\{\frac{\tau^{n}\,(1-u)^{2}}{(\tau-u)(1-\tau
u)}-1\right\\}$

with $g_{20V}(u,v)$ as in (2.22).

In particular, using Theorem 2.7 to identify the denominator of (5.20) as
$Z_{n}^{20V}$, this implies immediately the following:

######  Theorem 5.4.

The refined partition function generating function for the $20V$ model reads:

$Z_{n}^{20V}(\tau)=\det_{0\leq i,j\leq n-1}\left(g_{20V}^{\rm
ref}(u,v)|_{u^{i}v^{j}}\right)$

with $g_{20V}^{\rm ref}$ as in (5.22).

######  Remark 5.5.

We may interpret again the result as:

(5.23) $Z_{n}^{20V}(\tau)=\det\left((P^{(n)}(\tau))_{n}\right)$

the determinant of a finite truncation of an infinite matrix $P^{(n)}(\tau)$,
which differs from the matrix $P$ of Remark 2.8 only in its $n$-th column, and
is generated by $f_{P^{(n)}(\tau)}(u,v)=g_{20V}^{\rm ref}(u,v)$ (5.22).

Direct calculation using Theorem 5.4 gives the following first few values for
$n=1,2...,5$ of the refined partition function generating polynomials
$Z_{n}^{20V}(\tau)$ for the 20V model on the quadrangle $\mathcal{Q}_{n}$:

$\displaystyle Z_{1}^{20V}(\tau)$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle Z_{2}^{20V}(\tau)$ $\displaystyle=$ $\displaystyle
1+2\tau+\tau^{2}$ $\displaystyle Z_{3}^{20V}(\tau)$ $\displaystyle=$
$\displaystyle 4+15\tau+22\tau^{2}+15\tau^{3}+4\tau^{4}$ $\displaystyle
Z_{4}^{20V}(\tau)$ $\displaystyle=$ $\displaystyle
60+328\tau+772\tau^{2}+1008\tau^{3}+772\tau^{4}+328\tau^{5}+60\tau^{6}$
$\displaystyle Z_{5}^{20V}(\tau)$ $\displaystyle=$ $\displaystyle
3328+23868\tau+76856\tau^{2}+145860\tau^{3}+179088\tau^{4}+145860\tau^{5}$
$\displaystyle+76856\tau^{6}+23868\tau^{7}+3328\tau^{8}$

######  Remark 5.6.

Note that $Z_{n}^{20V}(\tau)$ is a palindromic polynomial of degree $2n-2$,
namely that $Z_{n,2n-k}^{20V}=Z_{n,k}^{20V}$ for all $k=1,2,...,2n-1$. Indeed,
from (5.22) we see that the only dependence on $\tau$ comes from the $n$-th
column of $P^{(n)}(\tau)$, generated by
$\varphi_{n}(\tau):=\left(\frac{1+u}{1-u}\right)^{2n}\frac{\tau^{n}\,(1-u)^{2}}{(\tau-u)(1-\tau
u)}$. The property follows from the fact that
$\varphi_{n}(\tau^{-1})=\tau^{-2(n-1)}\varphi_{n}(\tau)$.

Observe also that

(5.24) $Z_{n}^{20V}(0)=Z_{n,1}^{20V}=Z_{n,2n-1}^{20V}=Z_{n-1}^{20V}$

Indeed, the contributions to $Z_{n,2n-1}^{20V}$ all have a trivial topmost
path made of $n$ horizontal steps in the top row, followed by $2n-1$ down
steps along the rightmost vertical until its endpoint. This effectively cuts
out the top row and right column from the domain $\mathcal{Q}_{n}$, and the
remaining paths form a configuration on $\mathcal{Q}_{n-1}$. Similarly, the
contributions to $Z_{n,1}^{20V}=Z_{n}^{20V}(0)$ all have the bottom-most
diagonal entirely occupied by paths, while the rightmost vertical is empty
except for the bottom-most edge. This effectively cuts out the bottom diagonal
and right vertical lines of vertices, leaving us with paths on
$\mathcal{Q}_{n-1}$ as well.

Finally, Theorem 5.3 gives the corresponding $6V$ one-point functions:

$\displaystyle h_{1}^{6V}(\tau)$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle h_{2}^{6V}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1+\tau}{2}$ $\displaystyle h_{3}^{6V}(\tau)$
$\displaystyle=$ $\displaystyle\frac{4+7\tau+4\tau^{2}}{15}$ $\displaystyle
h_{4}^{6V}(\tau)$ $\displaystyle=$
$\displaystyle\frac{15+37\tau+37\tau^{2}+15\tau^{3}}{104}$ $\displaystyle
h_{5}^{6V}(\tau)$ $\displaystyle=$
$\displaystyle\frac{64+203\tau+282\tau^{2}+203\tau^{3}+64\tau^{4}}{816}$

We note again that these polynomials are palindromic, a property inherited
from the 20V one-point functions via the relation (5.12). In particular,
comparing the top and constant coefficients we get a non-trivial relation:
$Z_{n,1}^{6V}+Z_{n,2}^{6V}=Z_{n,2n-1}^{6V}$.

##  6\. Refined domino tilings of the Aztec triangle and connection to 20V
refined partition functions

###  6.1. Refined domino tiling partition functions

Figure 14. A typical configuration of non-intersecting Schröder paths for the
domino tiling model of the Aztec triangle $\mathcal{T}_{n}$, contributing to
the refined partition function $Z_{n,k}^{DT}$, i.e. such that the first entry
point of any path into the line $x+y=4n-2$ lies at some distance $k$ of the
endpoint $(2n-1,2n-1)$.

Consider again the path model for the domino tilings of the Aztec triangle
${\mathcal{T}}_{n}$. In any non-intersecting path configuration, the path
ending at the topmost point $(2n-1,2n-1)$ must first hit the diagonal line
$x+y=4n-2$ at some point $(2n-1-k,2n-1+k)$ for some $k\in[0,n-1]$. Let us
denote by $Z^{DT}_{n,k}$ the corresponding contribution to the total partition
function (see Fig. 14 for an illustration).

Alternatively, let us denote by $W^{DT}_{n,k}$ the partition function for non-
intersecting Schröder paths with starting points $(0,2i)$, $i=0,1,...,n-1$ and
endpoints $(2j+1,2j+1)$, $j=0,1,...,n-2$ together with the point
$(2n-1-k,2n-1+k)$. The condition of first hitting the line $y+x=4n-2$ at
$(2n-1-k,2n-1+k)$ is equivalent to imposing that the path visits the point
$(2n-1-k,2n-1+k)$ but does not visit the point $(2n-1-k-1,2n-1+k+1)$. Note
that once a point is visited on the line $x+y=4n-2$, the rest of the path
descends along this line until it reaches the endpoint $(2n-1,2n-1)$. This
implies immediately for all $k\in[0,n-1]$:

$Z^{DT}_{n,k}=W^{DT}_{n,k}-W^{DT}_{n,k+1}$

For convenience, we define the generating function:

$Z^{DT}(t):=\sum_{k=0}^{n-1}Z^{DT}_{n,k}\,t^{k}$

for refined domino tiling partition functions of the Aztec triangle
$\mathcal{T}_{n}$. We have the following:

######  Theorem 6.1.

The generating function $Z^{DT}(t)$ reads:

(6.1) $\displaystyle Z^{DT}(t)$ $\displaystyle=$ $\displaystyle\det_{0\leq
i,j\leq n-1}\left(f_{DT}(t;u,v)\Big{|}_{u^{i}v^{j}}\right)$ (6.2)
$\displaystyle f_{DT}^{\rm ref}(u,v)$ $\displaystyle=$ $\displaystyle
f_{DT}(u,v)+v^{n-1}\frac{{\alpha}_{+}(u)^{2n}}{\sqrt{1+6u+u^{2}}}\,\frac{(t-1)u}{{\alpha}_{+}(u)-tu}$

where ${\alpha}_{+}(u)=\frac{1}{2}\big{(}1+u+\sqrt{1+6u+u^{2}}\big{)}$ and
$f_{DT}(u,v)$ as in (3.5).

###### Proof.

Let us first compute $W^{DT}_{n,k}$. By direct application of the Lindström-
Gessel-Viennot theorem [Lin73, GV85], we have
$W^{DT}_{n,k}=\det(M_{n}^{(k)})$, where the $n\times n$ matrix $M_{n}^{(k)}$
differs from the matrix $M_{n}$ (of Lemma 3.1, Remark 3.2 and Theorem 3.3)
only in its last column, where the entry $M_{i,n-1}=\Sigma_{2n-1,2n-1-2i}$
(3.2) must be replaced by $(M_{n}^{(k)})_{i,n-1}=\Sigma_{2n-1-k,2n-1-2i+k}$ as
the corresponding endpoint has been shifted by $(-k,k)$. Recall that
$\Sigma_{i,j}=\sigma(u,v)|_{u^{\frac{i+j}{2}}v^{\frac{i-j}{2}}}$, with
$\sigma(u,v)$ as in (3.1). We deduce that

$\displaystyle\Sigma_{2n-1-k,2n-1-2i+k}$ $\displaystyle=$
$\displaystyle\oint\frac{dx}{2i\pi x}\frac{dy}{2i\pi
y}\frac{1}{x^{2n-1-i}y^{i-k}}\frac{1}{1-x-y-xy}$ $\displaystyle=$
$\displaystyle\oint\frac{du}{2i\pi u}\frac{dv}{2i\pi
v}\frac{1}{v^{2n-1-k}u^{i-k}}\frac{1}{1-v(1+u)-uv^{2}}$

by changing again to variables $u=y/x$, $v=x$. Noting that:

$\frac{1}{1-v(1+u)-uv^{2}}=\frac{1}{v\sqrt{1+6u+u^{2}}}\left(\frac{1}{1-v{\alpha}_{+}(u)}-\frac{1}{1-v{\alpha}_{-}(u)}\right)$

where

${\alpha}_{\pm}(u)=\frac{1}{2}\big{(}1+u\pm\sqrt{1+6u+u^{2}}\big{)}$

we get

$\frac{1}{1-v(1+u)-uv^{2}}\Bigg{|}_{v^{2n-1-k}}=\frac{{\alpha}_{+}(u)^{2n-k}-{\alpha}_{-}(u)^{2n-k}}{\sqrt{1+6u+u^{2}}}$

Note that ${\alpha}_{-}(u)=-u+O(u^{2})$ hence the term
${\alpha}_{-}(u)^{2n-k}=O(u^{2n-k})$ does not contribute to the coefficient of
$u^{i-k}$. The generating function for the $n$-th column of $M_{n}^{(k)}$
reads:

$v^{n-1}\sum_{i=0}^{n-1}\Sigma_{2n-1-k,2n-1-2i+k}\,u^{i}=v^{n-1}u^{k}\frac{{\alpha}_{+}(u)^{2n-k}}{\sqrt{1+6u+u^{2}}}+O(u^{n})$

The matrix $M_{n}^{(k)}$ is the finite truncation to the first $n$ rows and
columns of the infinite matrix $M^{(n,k)}$, with generating function

$f_{M^{(n,k)}}(u,v)=f_{M}(u,v)+v^{n-1}\frac{u^{k}\,{\alpha}_{+}(u)^{2n-k}-{\alpha}_{+}(u)^{2n}}{\sqrt{1+6u+u^{2}}}$

Using the linearity of the determinant w.r.t. its last column, we immediately
deduce that
$Z_{n,k}^{DT}=W^{DT}_{n,k}-W^{DT}_{n,k+1}=\det({\tilde{M}}_{n}^{(k)})$ where
the $n\times n$ matrix ${\tilde{M}}_{n}^{(k)}:=({\tilde{M}}^{(n,k)})_{n}$ is
the finite truncation of the infinite matrix ${\tilde{M}}^{(n,k)}$ with
generating function:

$f_{{\tilde{M}}^{(n,k)}}(u,v)=f_{M}(u,v)+v^{n-1}\frac{u^{k}{\alpha}_{+}(u)^{2n-k}-u^{k+1}{\alpha}_{+}(u)^{2n-k-1}-{\alpha}_{+}(u)^{2n}}{\sqrt{1+6u+u^{2}}}$

Similarly, the generating function
$Z^{DT}(t)=\det\Big{(}({\tilde{M}}^{(n)}(t))_{n}\Big{)}$ where the $n\times n$
matrix $({\tilde{M}}^{(n)}(t))_{n}$ is the finite truncation of the infinite
matrix ${\tilde{M}}^{(n)}(t)$ generated by:

$\displaystyle f_{{\tilde{M}}^{(n)}(t)}(u,v)$ $\displaystyle=$ $\displaystyle
f_{M}(u,v)+v^{n-1}\frac{\sum_{k\geq
0}t^{k}(u^{k}{\alpha}_{+}(u)^{2n-k}-u^{k+1}{\alpha}_{+}(u)^{2n-k-1})-{\alpha}_{+}(u)^{2n}}{\sqrt{1+6u+u^{2}}}$
$\displaystyle=$ $\displaystyle
f_{M}(u,v)+v^{n-1}\frac{{\alpha}_{+}(u)^{2n}}{\sqrt{1+6u+u^{2}}}\,\frac{(t-1)u}{{\alpha}_{+}(u)-tu}=f_{DT}^{\rm
ref}(u,v)$

(Here we have considered the sum over all $k\geq 0$, as the terms of order
$u^{n}$ and above do not affect the truncation.) This completes the proof of
Theorem 6.1. ∎

Direct calculation using Theorem 6.1 leads to the following first few values
of the refined partition function generating functions $Z_{n}^{DT}(t)$, up to
$n=5$:

$\displaystyle Z_{1}^{DT}(t)$ $\displaystyle=$ $\displaystyle 1$
$\displaystyle Z_{2}^{DT}(t)$ $\displaystyle=$ $\displaystyle 3+t$
$\displaystyle Z_{3}^{DT}(t)$ $\displaystyle=$ $\displaystyle 37+19t+4t^{2}$
$\displaystyle Z_{4}^{DT}(t)$ $\displaystyle=$ $\displaystyle
1780+1100t+388t^{2}+60t^{3}$ $\displaystyle Z_{5}^{DT}(t)$ $\displaystyle=$
$\displaystyle 324948+222716t+100724t^{2}+27196t^{3}+3328t^{4}$

We note the obvious sum rule $Z_{n}^{DT}(1)=Z_{n}^{DT}$. Observe also that the
leading coefficient of $t^{n-1}$ in $Z_{n}^{DT}(t)$ is nothing but

(6.3) $Z_{n-1,1}^{DT}=Z_{n-1}^{DT}$

Indeed, imposing that the topmost path (from $(0,2n-2)$ to $(2n-1,2n-1)$)
visits the topmost vertex on the line $x+y=4n-2$ forces this path to be a
trivial succession of $n$ up steps, followed by $n-1$ down steps, effectively
reducing the domain $\mathcal{T}_{n}$ to $\mathcal{T}_{n-1}$ for the $n-1$
remaining paths in each such configuration.

###  6.2. Relation between the refined tiling and refined 20V partition
functions

######  Theorem 6.2.

We have the following relation between refined partition functions of the
$20V$ model on the quadrangle $\mathcal{Q}_{n}$ and of the domino tilings of
the Aztec triangle $\mathcal{T}_{n}$:

(6.4)
$Z_{n}^{20V}(\tau)=\tau^{n}\,\frac{Z^{DT}(\tau)+\tau^{-1}Z^{DT}(\tau^{-1})}{1+\tau}$

###### Proof.

We apply the same technique as in the case of ordinary partition functions.
Let $L$ be the infinite matrix (4.1) of Sect. 4.2 and ${\tilde{M}}^{(n)}(t)$
generated by $f_{{\tilde{M}}^{(n)}(t)}(u,v)=f_{DT}^{\rm ref}(u,v)$ of Theorem
6.1. We compute:

(6.5) $\displaystyle f_{L{\tilde{M}}^{(n)}(t)}(u,v)$ $\displaystyle=$
$\displaystyle\frac{1+2u-u^{2}}{1-u}f_{{\tilde{M}}^{(n)}(t)}\left(u\frac{1+u}{1-u},v\right)=f_{P}(u,v)+v^{n-1}\left(\frac{1+u}{1-u}\right)^{2n}\frac{(t-1)u}{1-tu}$
$\displaystyle=$ $\displaystyle
g_{20V}(u,v)+v^{n-1}\left(\frac{1+u}{1-u}\right)^{2n}\left\\{\frac{1-u}{1-tu}-1\right\\}$

by use of the result $LM=P$ of Section 4.2. Let us now denote by
${\mathbf{C}}(t)$ the $n$-th column of the matrix $L{\tilde{M}}^{(n)}(t)$. We
have $Z^{DT}(t)=\det\left((L{\tilde{M}}^{(n)}(t))_{n}\right)$. By linearity of
the determinant w.r.t. the $n$-th column, the quantity
$\frac{t^{n}}{1+t}\big{(}Z^{DT}(t)+t^{-1}Z^{DT}(t^{-1})\big{)}$ is the
determinant of the finite truncation of an infinite matrix
${\bar{M}}^{(n)}(t)$ defined as follows: it is identical to
$L{\tilde{M}}^{(n)}(t)$ except for the $n$-th column, where ${\mathbf{C}}(t)$
is replaced with the combination
$\frac{t^{n}}{1+t}\\{{\mathbf{C}}(t)+t^{-1}{\mathbf{C}}(t^{-1})\\}$.
Consequently:

$\displaystyle t^{n}\,\frac{Z^{DT}(t)+t^{-1}Z^{DT}(t^{-1})}{t+1}$
$\displaystyle=$ $\displaystyle\det\left(({\bar{M}}^{(n)}(t))_{n}\right)$
$\displaystyle f_{{\bar{M}}^{(n)}(t)}(u,v)$ $\displaystyle=$ $\displaystyle
g_{20V}(u,v)+v^{n-1}\left(\frac{1+u}{1-u}\right)^{2n}\left\\{t^{n}\,\frac{\frac{1-u}{1-tu}+\frac{1-u}{t-u}}{1+t}-1\right\\}$
$\displaystyle=$ $\displaystyle g_{20V}^{\rm ref}(u,v)$

where we have recognized the generating function $g^{\rm ref}_{20V}(u,v)$ (at
$\tau=t$) of Theorem 5.3. The theorem follows by setting $t=\tau$, and using
the identification ${\bar{M}}^{(n)}(\tau)=P^{(n)}(\tau)$ given by (5.23). ∎

######  Remark 6.3.

Note that the result above gives an independent confirmation that the
polynomial $Z_{n}^{20V}(\tau)$ is palindromic, namely that
$\tau^{2n-2}Z_{n}^{20V}(\tau^{-1})=Z_{n}^{20V}(\tau)$ as noted in Remark 5.6.

Finally Eqn.(6.4) implies the following relations between the refined
partition functions of the $20V$ model and the domino tiling problem.

######  Corollary 6.4.

The refined partition functions for the $20V$ model and for the domino tilings
of the Aztec triangle are related as follows:

(6.6)
$Z_{n,k}^{DT}=Z_{n,n+k+1}^{20V}+Z_{n,n+k}^{20V}=Z_{n,n-k-1}^{20V}+Z_{n,n-k}^{20V}$

for $k=0,1,...,n-1$ with the convention that $Z_{n,0}^{20V}=Z_{n,2n}^{20V}=0$.

##  7\. Pentagonal 20V and truncated Aztec triangle domino tilings

In this section, we prove the correspondence (20V DWBC3-DT conjecture of
[DFG20]) between 20V configurations on the domains $\mathcal{P}_{n,n-k}$ (see
Fig. 4) and the domino tiling configurations of the domains
$\mathcal{T}_{n,n-k}$ for $k=2$ and $3$. The method of proof is purely
combinatorial and does not rely on integrability, except for its use of the
$k=1$ results.

###  7.1. Proof of the 20V-DT conjecture for $k=2$

Figure 15. The proof of the 20V-DT conjecture for $k=2$. The 20V partition
function on $\mathcal{Q}_{n}$ is decomposed according to the two possible
configurations of the bottom vertex (a) and (b). The domino tiling partition
function on $\mathcal{T}_{n}$ is similarly decomposed according to whether the
topmost vertex is visited (c) or not (d).

######  Theorem 7.1.

The number of configurations of the 20V DWBC3 model on the pentagon
$\mathcal{P}_{n,n-2}$ is identical to the number of domino tilings of the
truncated Aztec triangle $\mathcal{T}_{n,n-2}$, namely:

$Z^{20V}({\mathcal{P}}_{n,n-2})=Z^{DT}({\mathcal{T}}_{n,n-2})$

###### Proof.

The correspondence is proved by expressing both relevant partition function in
terms of known objects. For the 20V model on $\mathcal{P}_{n,n-2}$, let us
start from the partition function $Z_{n}^{20V}$ on the larger domain
$\mathcal{Q}_{n}=\mathcal{P}_{n,n-1}$. This quantity splits into two parts,
according to the two possible local configuration of the bottom-most vertex
displayed in Figs.15 (a) and (b) (circled vertex). In the case (a), the
configuration of the bottom vertex induces a chain of forced edge occupations,
due to the SW boundary condition. The contribution of case (a) is readily seen
to reduce to $Z_{n,1}^{20V}=Z_{n-1}^{20V}$. The case (b) contributes exactly
$Z^{20V}({\mathcal{P}}_{n,n-2})$. We deduce the relation:

$Z^{20V}({\mathcal{P}}_{n,n-2})=Z^{20V}_{n}-Z^{20V}_{n-1}$

For the domino tiling of $\mathcal{T}_{n,n-2}$ let us also start from the
partition function $Z_{n}^{DT}$ on the larger domain
$\mathcal{T}_{n}=\mathcal{T}_{n,n-1}$ in the lattice path formulation.
Analogously, this quantity splits into two parts according to whether the
topmost vertex is visited or not, as displayed in Figs.15 (c) and (d). The
case (c) corresponds to a unique configuration of the topmost path, and the
contribution reduces to $Z_{n-1}^{DT}$. The case (d) contributes precisely
$Z^{DT}({\mathcal{T}}_{n,n-2})$. This gives the relation

$Z^{DT}({\mathcal{T}}_{n,n-2})=Z^{DT}_{n}-Z^{DT}_{n-1}$

The theorem follows from the result of Theorem 4.3, i.e. the identities
$Z^{20V}_{m}=Z^{DT}_{m}$ for all $m$. ∎

###  7.2. Proof of the 20V-DT conjecture for $k=3$

Figure 16. Decomposition of $Z^{20V}({\mathcal{P}}_{n,n-2})$ into four parts
(a), (c),(d), (e), according to the bottom vertex configurations (circled).
Cases (a) are further decomposed according to the point of exit of the
rightmost path at height $k$ (empty circle) from the second rightmost vertical
line. There are $2$ distinct configurations of the rightmost path after this
exit point (paths using edges among the dashed ones), in $2$-1 correspondence
with the partition function (b) on $\mathcal{Q}_{n-1}$, equal to the sum of
the refined partition function $\sum_{j\geq k}Z_{n,j}^{20V}$. Cases (c) and
(d) sum up to $Z_{n,2}^{20V}-Z_{n,1}^{20V}$, while (e) contributes
$Z^{20V}({\mathcal{P}}_{n,n-3})$. Figure 17. Decomposition of
$Z^{DT}({\mathcal{P}}_{n,n-2})$ into three pieces (a), (b), (c) according to
the configurations of the two top vertices (empty circles). Case (a) gives
rise to $2(n-1)$ times the domino problem on $\mathcal{T}_{n-1}$, as there are
$2(n-1)$ choices for how the top path ends, using only edges among the dashed
ones. Case (b) receives contributions from the refined partition function
where paths first enter the $x+y=4n-2$ line at distance $n-1$ from the top
endpoint (i.e. at the right top vertex), but that do not visit the left
vertex. Case (c) contributes $Z^{DT}({\mathcal{T}}_{n,n-3})$.

######  Theorem 7.2.

The number of configurations of the 20V DWBC3 model on the pentagon
$\mathcal{P}_{n,n-3}$ is identical to the number of domino tilings of the
truncated Aztec triangle $\mathcal{T}_{n,n-3}$, namely:

$Z^{20V}({\mathcal{P}}_{n,n-3})=Z^{DT}({\mathcal{T}}_{n,n-3})$

###### Proof.

We proceed like in the case $k=2$. For the 20V model on $\mathcal{P}_{n,n-3}$,
let us start from the partition function $Z^{20V}({\mathcal{P}}_{n,n-2})$. It
splits into four parts according to the local configuration of the two bottom
vertices, as shown in Figs.16 (a), (c), (d) and (e).

The first contribution (a) can be further split according to the configuration
of the rightmost column: there are two ways in which the path can leave the
second rightmost vertical line at an exit point say at height $k$ (by a
horizontal or diagonal step, followed by vertical steps until the endpoint).
Removing the last column, and inserting vertical steps up to the point of exit
at height $k$ into the second vertical, we get a 2 to 1 mapping to refined
configurations of the 20V on $\mathcal{Q}_{n-1}$ that reach the last column at
or above height $k$, resulting in a contribution $\sum_{j\geq
k}Z_{n-1,j}^{20V}$ for each $k$, hence a total of
$2\sum_{k=1}^{2n-3}\sum_{j\geq
k}Z_{n-1,j}^{20V}=\sum_{k=1}^{2n-3}2k\,Z_{n-1,k}^{20V}$.

The case (c) corresponds to the contributions to the refined partition
function $Z_{n,2}^{20V\backslash}$ such that a path goes vertically through
the left vertex. We must therefore subtract from $Z_{n,2}^{20V\backslash}$ the
contributions with the two rightmost paths ending diagonally before their last
vertical step, in bijection with 20V configurations on $\mathcal{Q}_{n-1}$.
This gives a net contribution of $Z_{n,2}^{20V\backslash}-Z_{n-1}^{20V}$. The
case (d) contributes $Z_{n,2}^{20V-}$. Finally the case (e) contributes
$Z^{20V}({\mathcal{P}}_{n,n-3})$. Collecting all the terms and using
$Z_{n,2}^{20V\backslash}+Z_{n,2}^{20V-}=Z_{n,2}^{20V}$, we get:

$Z^{20V}({\mathcal{P}}_{n,n-2})=\sum_{k=1}^{2n-3}2k\,Z_{n-1,k}^{20V}+Z_{n,2}^{20V}-Z_{n-1}^{20V}+Z^{20V}({\mathcal{P}}_{n,n-3})$

Using Remarks 5.6 and 6.3, which imply the symmetry
$Z_{n-1,2n-2-k}^{20V}=Z_{n-1,k}^{20V}$, we further note that

$\sum_{k=1}^{2n-3}2k\,Z_{n-1,k}^{20V}=\frac{1}{2}\sum_{k=1}^{2n-3}2k(Z_{n-1,k}^{20V}+Z_{n-1,2n-2-k}^{20V})=2(n-1)\sum_{k=1}^{2n-3}Z_{n-1,k}^{20V}=2(n-1)Z_{n-1}^{20V}$

leading to the identity:

(7.1)
$Z^{20V}({\mathcal{P}}_{n,n-3})=Z^{20V}_{n}-Z^{20V}_{n,2}-2(n-1)Z^{20V}_{n-1}$

For the domino tiling model, we start from $Z^{DT}({\mathcal{T}}_{n,n-2})$. It
splits again according to the configurations of it two topmost vertices, as
shown in Figs.17 (a) and (b). The case (a) corresponds to a path visiting the
leftmost vertex, which must be made of $n-1$ first up steps, and continue on
the set of dashed edges until it reaches the endpoint $(2n-1,2n-1)$, giving
rise to $2(n-1)$ possibilities, while the rest of the configuration is
arbitrary on $\mathcal{Q}_{n-1}$. The corresponding contribution is
$2(n-1)Z_{n-1}^{DT}$. In the case (b) a path visits the right vertex, while
the left vertex is empty. The corresponding configurations are those paths
contributing to the refined partition function $Z_{n,n-2}^{DT}$ that do not
pass through the left vertex. We must therefore subtract from $Z_{n,n-2}^{DT}$
the two contributions of the topmost path that visit both vertices in
$\mathcal{T}_{n,n-2}$ (with either a horizontal step between the two vertices,
or a down followed by an up step). In both cases, the rest of the paths
correspond to an arbitrary tiling of $\mathcal{T}_{n-1}$. The total
contribution is therefore $Z_{n,n-2}^{DT}-2Z_{n,n-1}^{DT}$. The case (c) with
the two empty vertices contributes $Z^{DT}({\mathcal{T}}_{n,n-3})$. Assembling
all the terms, we get

$Z^{DT}({\mathcal{T}}_{n,n-2})=2(n-1)Z_{n-1}^{DT}+Z_{n,n-2}^{DT}-2Z_{n,n-1}^{DT}+Z^{DT}({\mathcal{T}}_{n,n-3})$

We deduce that

(7.2)
$Z^{DT}({\mathcal{T}}_{n,n-3})=Z^{DT}_{n}-Z_{n,n-2}^{DT}-(2n-3)Z^{DT}_{n-1}$

Finally we use eq.(6.6) of Corollary 6.4 for $k=n-2$ to identify:
$Z_{n,n-2}^{DT}=Z_{n,1}^{20V}+Z_{n,2}^{20V}$. Together with the identities
$Z^{20V}_{m}=Z^{DT}_{m}$ of Theorem 4.3 and the fact that
$Z_{n,1}^{20V}=Z_{n-1}^{20V}$ (5.24), this implies the equality between the
r.h.s. of (7.1) and (7.2), and the theorem follows. ∎

##  8\. Discussion and Conclusion

In this paper we have investigated the 20V model with DWBC3 boundary
conditions on the family of pentagons $\mathcal{P}_{n,n-k}$ and their relation
to the domino tilings of the family of Aztec triangles $\mathcal{T}_{n,n-k}$,
and proved the conjectured identity between their partition functions for the
cases $k=1$, $2$, $3$. In the case $k=1$ of the quadrangle
$\mathcal{Q}_{n}=\mathcal{P}_{n,n-1}$ we have extended the result to refined
partition functions. Our proofs for this case have relied mainly on the choice
of integrable weights for the 20V model, allowing for transforming the
original problem into a 6V model with U-turn boundaries. The further
truncations of the domain $\mathcal{Q}_{n}$ for $k=2,3$ were treated in a
purely combinatorial manner, by relating the partition functions to the $k=1$
case and its refinements.

We now discuss a conjecture for the exact number $Z_{n}^{20V}=Z_{n}^{DT}$ and
alternative formulas for these quantities. We also show how to introduce some
extra weight in the domino tiling problems, before making some concluding
remarks.

###  8.1. An exact (conjectured) formula

Based on numerical data, we were able to formulate the following conjecture
about the total number of configurations of the 20V model on the quadrangle
$\mathcal{Q}_{n}$:

######  Conjecture 8.1.

The total numbers $Z^{20V}_{n}=Z_{n}^{DT}$ of configurations of the 20V model
on the quadrangle $\mathcal{Q}_{n}$ and of domino tilings of the Aztec
triangle $\mathcal{T}_{n}$ read:

(8.1)
$Z^{20V}_{n}=Z_{n}^{DT}=2^{n(n-1)/2}\prod_{i=0}^{n-1}\frac{(4i+2)!}{(n+2i+1)!}$

This formula was checked up to $n=30$. It is reminiscent of the famous formula
for the number $A_{n}$ of $n\times n$ Alternating Sign Matrices (ASM), which
reads $A_{n}=\prod_{i=0}^{n-1}\frac{(3i+1)!}{(n+i)!}$ or the related formula
for the dimension $\delta_{n}=3^{n(n-1)/2}\,A_{n}$ of the irreducible
representation of $GL_{n}$ indexed by the Young diagram
$Y_{n}=(n-1,n-1,n-2,n-2,...,1,1,0,0)$.

Note that the technique used by Kuperberg [Kup96, Kup02] for proving the ASM
formula and its variants doesn’t seem to apply here, as we have not been able
to find a factorized formula for the partition function in which the spectral
parameters approach their combinatorial value as
$z_{i}=q^{4}s^{i},w_{j}=q^{-8}s^{j+n}$ for some parameter $s$ to be sent to
$1$ eventually.

Another approach would try to reproduce the connection between the 6V model
with DWBC conditions and the $GL_{n}$ character with partition $Y_{n}$ or that
between U-turn 6V model and characters of $SP_{2n}$ [Str06, RS04b]. Looking
for candidates, and denoting by $B_{n}$ the r.h.s. of (8.1), we have
found555All these results are product formulas obtained by specializing the
corresponding characters, given by various determinant formulas [Miz03], when
all arguments are $1$. that the irreducible representation of $SP_{6n}$ with
diagram $Y_{n}^{\prime}=(n-1,n-1,n-1,n-2,n-2,n-2,...,1,1,1,0,0,0)$ has the
dimension $\delta_{n}^{\prime}=B_{2n}/2^{2n}$. Analogously, we found that the
irreducible representation of $SO_{6n+1}$ with diagram
$Y_{n}^{\prime\prime}=(\frac{2n-1}{2},\frac{2n-1}{2},\frac{2n-1}{2},...,\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2})$
has dimension $\delta_{n}^{\prime\prime}=2^{n}\,B_{2n}$. Finally, we found
that the irreducible representation of $SO_{6n-2}$ with diagram
$Y_{n}^{\prime\prime\prime}=(n-1,n-1,n-1,n-2,n-2,n-2,...,1,1,1,0,0)$ has
dimension $\delta_{n}^{\prime\prime\prime}=2^{n-1}\,B_{2n-1}$. Recalling that
the in 6V case [Str06, RS04b] the argument involved some non-trivial
${\mathbb{Z}}_{3}$ symmetry of the partition function at the combinatorial
point, we expect a ${\mathbb{Z}}_{4}$ symmetry to play an analogous role here.
We intend to return to this question in a future publication.

###  8.2. Alternative expressions for $Z_{n}^{20V}$

The results of this paper allow us to derive several alternative determinant
formulas for the number $Z_{n}^{20V}$. Using Theorem 2.7, we may express
$Z_{n}^{20V}$ as follows.

######  Theorem 8.2.

The number $Z_{n}^{20V}$ of configurations of the $20V$ model on the
quadrangle $\mathcal{Q}_{n}$ reads:

(8.2) $Z_{n}^{20V}=\det_{0\leq i\leq j\leq n-1}\left(2^{i}\,{i+2j+1\choose
2j+1}-{i-1\choose 2j+1}\right)$

with the convention ${m\choose p}=0$ for all $-1\leq m<p$.

###### Proof.

We start from the determinant formula $Z_{n}^{20V}=\det_{0\leq i,j\leq
n-1}(f_{P}(u,v)|_{u^{i}v^{j}})=\det(P_{n})$ of Theorem 2.7, where:

$f_{P}(u,v)=g_{20V}(u,v)=\frac{\left(\frac{1+u}{1-u}\right)^{2}}{1-v\,\left(\frac{1+u}{1-u}\right)^{2}}-\frac{u^{2}}{1-v\,u^{2}}$

and $P_{n}$ is the finite truncation of the infinite matrix $P$ generated by
$f_{P}$. Using the same technique as in Sections 4.1-4.2, let us consider the
infinite matrix $\Lambda$ with generating function:

$f_{\Lambda}(u,v)=\frac{1-u}{1-u-vu}$

Then (1) $\Lambda$ is lower triangular and (2) the diagonal matrix elements of
$\Lambda$ are all $1$. We deduce the truncation
$(\Lambda\,P)_{n}=\Lambda_{n}\,P_{n}$, and therefore
$\det((\Lambda\,P)_{n})=\det(P_{n})$. Explicitly computing:

$f_{\Lambda\,P}(u,v)=(f_{\Lambda}*f_{P})(u,v)=f_{P}\left(\frac{u}{1-u},v\right)=\frac{\left(\frac{1}{1-2u}\right)^{2}}{1-v\,\left(\frac{1}{1-2u}\right)^{2}}-\frac{\left(\frac{u}{1-u}\right)^{2}}{1-v\,\left(\frac{u}{1-u}\right)^{2}}$

we identify the entries

$(\Lambda
P)_{i,j}=\left\\{\left(\frac{1}{1-2u}\right)^{2j+2}-\left(\frac{u}{1-u}\right)^{2j+2}\right\\}\Bigg{|}_{u^{i}}=2^{i}\,{i+2j+1\choose
2j+1}-{i-1\choose 2j+1}$

and the theorem follows. ∎

A slight variant of the formula consists of using the analytic continuation of
binomial coefficients.

######  Corollary 8.3.

The number $Z_{n}^{20V}$ of configurations of the $20V$ model on the
quadrangle $\mathcal{Q}_{n}$ reads:

(8.3) $Z_{n}^{20V}=\frac{2^{n(n-1)/4}}{2}\,\det_{0\leq i,j\leq
n-1}\Big{(}\theta_{2j+1}(i)+\theta_{2j+1}(-i)\Big{)},\quad\theta_{m}(x)=\frac{2^{x/2}}{m!}(x+1)(x+2)...(x+m)$

This latter, more symmetric form allows to derive the following constant term
identity.

######  Theorem 8.4.

The number $Z_{n}^{20V}$ of configurations of the $20V$ model on the
quadrangle $\mathcal{Q}_{n}$ reads:

$Z_{n}^{20V}=CT_{x_{1},...,x_{n}}\left\\{\frac{\prod_{1\leq i<j\leq
n}(x_{i}-x_{j})(1+x_{i}+x_{j}-x_{i}x_{j})}{\prod_{i=1}^{n}x_{i}^{2i-1}\,(1-x_{i})^{n}}\right\\}$

where the symbol $CT_{x_{1},...,x_{n}}$ stands for the constant coefficient in
the Laurent series expansion around $x_{i}=0$ for all $i$.

###### Proof.

Let us rewrite the function $\theta_{m}(x)$ of (8.3) as:

$\theta_{m}(x)=\frac{1}{m!}\partial_{u}^{m}\,u^{m+x}\big{|}_{u=\sqrt{2}}=CT_{t}\left\\{e^{t\,\partial_{u}}t^{-m}\,u^{m+x}\right\\}\big{|}_{u=\sqrt{2}}=CT_{t}\left\\{t^{-m}\,(\sqrt{2}+t)^{m+x}\right\\}$

where we have used the Taylor expansion to interpret the operator
$e^{t\,\partial_{u}}$ as the shift map $S_{t}:f(u)\mapsto f(u+t)$. We now use
multilinearity of determinants to rewrite

$Z_{n}^{20V}=\frac{2^{n(n-1)/4}}{2}\,CT_{t_{1},t_{2},...,t_{n}}\left\\{\prod_{j=1}^{n}\left(1+\frac{\sqrt{2}}{t_{j}}\right)^{2j-1}\,\det_{1\leq
i,j\leq n}\left((\sqrt{2}+t_{j})^{i-1}+(\sqrt{2}+t_{j})^{1-i}\right)\right\\}$

Next we use the standard formula

$\det_{1\leq i,j\leq n}(x_{j}^{i-1}+x_{j}^{1-i})=2\,\frac{\prod_{1\leq i<j\leq
n}(x_{i}-x_{j})(1-x_{i}x_{j})}{\prod_{i=1}^{n}x_{i}^{n-1}}$

to rewrite:

(8.4)
$Z_{n}^{20V}=2^{n(n-1)/4}\,CT_{t_{1},t_{2},...,t_{n}}\left\\{\prod_{i=1}^{n}\left(1+\frac{\sqrt{2}}{t_{i}}\right)^{2i-1}\,\prod_{1\leq
i<j\leq
n}(t_{j}-t_{i})\left(1-\frac{1}{(\sqrt{2}+t_{i})(\sqrt{2}+t_{j})}\right)\right\\}$

We now use the obvious property of the constant term that
$CT_{\mathbf{x}}(f({\mathbf{x}}))=CT_{\mathbf{x}}({\rm Sym}(f({\mathbf{x}}))$,
where for any function of the variables ${\mathbf{x}}=(x_{1},...,x_{n})$ we
define the symmetrization map

${\rm Sym}:f({\mathbf{x}})\mapsto{\rm
Sym}(f)({\mathbf{x}}):=\frac{1}{n!}\sum_{\sigma\in
S_{n}}f(x_{\sigma(1)},...,x_{\sigma(n)}).$

We also need the antisymmetrization map

${\rm ASym}:f({\mathbf{x}})\mapsto{\rm
ASym}(f)({\mathbf{x}}):=\frac{1}{n!}\sum_{\sigma\in S_{n}}{\rm
sgn}(\sigma)\,f(x_{\sigma(1)},...,x_{\sigma(n)})$

with the main property

${\rm Sym}\left(\prod_{1\leq i<j\leq
n}(x_{j}-x_{i})f(x_{1},...,x_{n})\right)=\prod_{1\leq i<j\leq
n}(x_{j}-x_{i})\,{\rm ASym}\left(f(x_{1},...,x_{n})\right).$

Symmetrizing the argument of the constant term in (8.4), we get:

$\displaystyle Z_{n}^{20V}$ $\displaystyle=$
$\displaystyle\frac{2^{n(n-1)/2}}{n!}CT_{t_{1},t_{2},...,t_{n}}\left\\{\prod_{i=1}^{n}\left(1+\frac{\sqrt{2}}{t_{i}}\right)\right.$
$\displaystyle\quad\times\,\left.\prod_{1\leq i<j\leq
n}\left(1-\frac{1}{(\sqrt{2}+t_{i})(\sqrt{2}+t_{j})}\right)(t_{j}-t_{i})\left(\left(1+\frac{\sqrt{2}}{t_{j}}\right)^{2}-\left(1+\frac{\sqrt{2}}{t_{i}}\right)^{2}\right)\right\\}.$

Let us now change variables to $x_{i}=\frac{t_{i}}{t_{i}+\sqrt{2}}$. Viewing
the constant term in $t_{j}$ as a contour integral $\oint\frac{dt_{j}}{2\pi
i\,t_{j}}=\oint\frac{dx_{j}}{2\pi i\,x_{j}(1-x_{j})}$, we easily find:

$\displaystyle Z_{n}^{20V}$ $\displaystyle=$
$\displaystyle\frac{1}{n!}CT_{x_{1},x_{2},...,x_{n}}\left\\{\frac{\prod_{1\leq
i<j\leq
n}(1+x_{i}+x_{j}-x_{i}x_{j})(x_{i}-x_{j})(x_{j}^{-2}-x_{i}^{-2})}{\prod_{i=1}^{n}x_{i}\,(1-x_{i})^{n}}\right\\}$
$\displaystyle=$ $\displaystyle CT_{x_{1},x_{2},...,x_{n}}\left\\{{\rm
Sym}\left(\frac{\prod_{1\leq i<j\leq
n}(1+x_{i}+x_{j}-x_{i}x_{j})(x_{i}-x_{j})}{\prod_{i=1}^{n}x_{i}^{2i-1}\,(1-x_{i})^{n}}\right)\right\\}.$

The theorem follows. ∎

The constant term identity of Theorem 8.4 is reminiscent of constant term
expressions for TSSCPP derived by Zeilberger [Zei96] in his famous proof of
the ASM conjecture.

###  8.3. Introducing step weights in the domino tiling problem

In this paper we have found compact determinant formulas for both the number
of tilings of the Aztec triangle $\mathcal{T}_{n}$ and its refinements,
involving the truncation of infinite matrices generated respectively by the
series $f_{DT}$ (3.5) and $f_{DT}^{\rm ref}$ (6.2). We note that a very simple
decoration of the generating function $f_{DT}(u,v)$ allows to include a non-
trivial extra step weight for the non-intersecting Schröder paths. In general,
one would want to associate weights ${\alpha},\beta,\gamma$ respectively to
up, down and horizontal steps. However, by simple rescaling of $u,v$ we may
restrict without loss of generality to only a weight $\gamma$ per horizontal
step.

######  Theorem 8.5.

The partition function for domino tilings of the Aztec triangle of order $n$,
with an extra weight $\gamma$ per horizontal step in the non-intersecting
Schröder path formulation reads:

(8.5) $\displaystyle Z^{DT,\gamma}_{n}$ $\displaystyle=$
$\displaystyle\det_{0\leq i,j\leq
n-1}\left(f_{DT,\gamma}(u,v)\Big{|}_{u^{i}v^{j}}\right)$ (8.6) $\displaystyle
f_{DT,\gamma}(u,v)$ $\displaystyle=$
$\displaystyle\frac{1+u}{1-v-2(1+\gamma)uv-u^{2}v+\gamma^{2}u^{2}v^{2}}$

###### Proof.

We adapt the proof of Theorem 3.3. By the Lindström-Gessel-Viennot theorem,
the partition function function of the non-intersecting Schröder paths with
the new weights is given by $Z^{DT,\gamma}_{n}=\det((M_{\gamma})_{n})$, where
the subscript $n$ indicates the truncation to the first $n$ rows and columns
of the infinite matrix $M_{\gamma}$ whose entries $(M_{\gamma})_{i,j}$ are the
partition functions for a single path form $(0,2i)$ to $(2j+1,2j+1)$, counted
with a multiplicative weight $\gamma$ per horizontal step. We simply have to
show that these entries are generated by:

$f_{M_{\gamma}}(u,v)=\frac{1+u}{1-v-2(1+\gamma)uv-u^{2}v+\gamma^{2}u^{2}v^{2}}=f_{DT,\gamma}(u,v)$

To show this, we start with the function
$\sigma_{\gamma}(u,v):=\frac{1}{1-u-v-\gamma uv}$ the partition function for
arbitrary finite length Schröder paths from the origin, with weight $u$ per up
step, $v$ per down step, and $\gamma uv$ per horizontal step. Accordingly the
partition function $(\Sigma_{\gamma})_{i,j}$ of paths from the origin with
fixed endpoint $(i,j)$, with $i-j$ even, is given by
$(\Sigma_{\gamma})_{i,j}=\sigma_{\gamma}(u,v)|_{u^{\frac{i+j}{2}}v^{\frac{i-j}{2}}}$,
or equivalently by a straightforward generalization of (3.2). Noting again
that $(M_{\gamma})_{i,j}=(\Sigma_{\gamma})_{2j+1,2j+1-2i}$, we get

$(M_{\gamma})_{i,j}=\oint\frac{dx}{2i\pi x}\frac{dy}{2i\pi
y}\,\frac{1}{x^{2j+1-i}y^{i}}\,\frac{1}{1-x-y-\gamma xy}=\oint\frac{du}{2i\pi
u}\frac{dv}{2i\pi v}\,\frac{1}{v^{2j+1}u^{i}}\,\frac{1}{1-v-uv-\gamma uv^{2}}$

after again changing integration variables to $u=y/x,v=x$. Using the same
trick (3.3) as in previous section, we conclude that:

$f_{M_{\gamma}}(u,v)=\frac{\frac{1}{1-(1+u)\sqrt{v}-\gamma
uv}-\frac{1}{1+(1+u)\sqrt{v}-\gamma uv}}{2\sqrt{v}}=\frac{1+u}{(1-\gamma
uv)^{2}-v(1+u)^{2}}$

and the Theorem follows. ∎

The result above is easily extended to the refined domino tiling of Section.
6. The partition function
$Z_{n}^{DT,\gamma}(t):=\sum_{k=1}^{n}Z^{DT,\gamma}_{n,k}\,t^{k-1}$ now
including a multiplicative weight $\gamma$ per horizontal step in the non-
intersecting Schröder path formulation of the refined tiling problem, can be
directly calculated as follows (we leave the proof to the reader as a
straightforward exercise).

######  Theorem 8.6.

The generating function $Z^{DT}(t,\gamma)$ reads:

$\displaystyle Z^{DT}(t,\gamma)$ $\displaystyle=$ $\displaystyle\det_{0\leq
i,j\leq n-1}\left(f_{DT,\gamma}^{\rm ref}(u,v)\Big{|}_{u^{i}v^{j}}\right)$
$\displaystyle f_{DT,\gamma}^{\rm ref}(u,v)$ $\displaystyle=$ $\displaystyle
f_{DT,\gamma}(u,v)+v^{n-1}\frac{{\alpha}_{+}(\gamma;u)^{2n}}{\sqrt{1+2(1+2\gamma)u+u^{2}}}\,\frac{(t-1)u}{{\alpha}_{+}(\gamma;u)-tu}$

where
${\alpha}_{+}(\gamma,u)=\frac{1}{2}\big{(}1+u+\sqrt{1+2(1+2\gamma)u+u^{2}}\big{)}$
and $f_{DT,\gamma}(u,v)$ as in (8.6).

It is interesting to notice that the upper triangular matrix $L$ involved in
the proofs of Theorems 4.3 and 6.2 can be modified to obtain an alternative
expression for the partition function generating function $Z^{DT,\gamma}(t)$
above. Indeed, let $L_{\gamma}$ be the infinite matrix generated by:

$f_{L_{\gamma}}(u,v):=\frac{1+2\gamma u-\gamma u^{2}}{1-u-vu(1+\gamma u)}$

Writing
$Z^{DT}(t,\gamma)=\det\big{(}(L_{\gamma}{\tilde{M}}_{\gamma}^{(n)}(t))_{n}\big{)}$,
where ${\tilde{M}}_{\gamma}^{(n)}(t)$ is generated by $f_{DT,\gamma}^{\rm
ref}(u,v)$ above, we finally get:

######  Theorem 8.7.

The generating function $Z^{DT}(t,\gamma)$ reads:

$\displaystyle Z^{DT}(t,\gamma)$ $\displaystyle=$
$\displaystyle\det\Big{(}{\bar{f}}_{DT,\gamma}^{\rm
ref}(u,v)\big{|}_{u^{i}v^{j}}\Big{)}$ $\displaystyle{\bar{f}}_{DT,\gamma}^{\rm
ref}(u,v)$ $\displaystyle=$ $\displaystyle\frac{(1+\gamma u^{2})(1+2\gamma
u-\gamma u^{2})}{(1-\gamma^{2}u^{2}v)((1-u)^{2}-v(1+\gamma
u)^{2})}+v^{n-1}\left(\frac{1+\gamma u}{1-u}\right)^{2n}\frac{(t-1)u}{1-tu}$

###### Proof.

We use the infinite matrix
${\bar{M}}_{\gamma}^{(n)}(t):=L_{\gamma}{\tilde{M}}_{\gamma}^{(n)}(t)$, whose
generating function is easily computed via the formula
$f_{L_{\gamma}\,{\tilde{M}}_{\gamma}^{(n)}(t)}(u,v)=\frac{1+2\gamma u-\gamma
u^{2}}{1-u}f_{DT,\gamma}^{\rm ref}\left(u\frac{1+\gamma
u}{1-u},v\right)={\bar{f}}_{DT,\gamma}^{\rm ref}(u,v)$. ∎

Note also that

$\frac{(1+\gamma u^{2})(1+2\gamma u-\gamma
u^{2})}{(1-\gamma^{2}u^{2}v)((1-u)^{2}-v(1+\gamma
u)^{2})}=\frac{\left(\frac{1+\gamma
u}{1-u}\right)^{2}}{1-v\left(\frac{1+\gamma
u}{1-u}\right)^{2}}-\frac{\gamma^{2}u^{2}}{1-v\gamma^{2}u^{2}}$

This $\gamma$-deformation of (2.22) is very suggestive, however we have not
been able to find a counterpart of the parameter $\gamma$ in the 20V model.

###  8.4. Conclusion

Despite the great progress on proving the conjectures of Ref. [DFG20], we are
still only understanding part of the sequence of correspondences between the
20V model on the domains $\mathcal{P}_{n,k}$ and the domino tiling of the
domains $\mathcal{T}_{n,k}$. One way to gain a better global understanding of
these would be to include deformation (spectral?) parameters in the
correspondence of partition functions on $\mathcal{Q}_{n}$ and
$\mathcal{T}_{n}$ respectively. However, the number of deformation parameters
at hand is limited. For instance, the trick we used with the last $w_{n}=w$
vertical spectral parameter cannot be adapted to horizontal spectral
parameters: these are indeed all frozen to the same value $z=p^{-2}q^{-2}$ in
order for the relation to the U-turn boundary 6V partition function to hold.
One may however try to keep more non-trivial values of the vertical spectral
parameters $w_{n},w_{n-1},...$ etc.

Another direction of study regards large $n$ asymptotics. We expect the
typical large configurations of the 20V model on $\mathcal{Q}_{n}$ to exhibit
an arctic curve phenomenon, such as that observed in [DDFG20] for the 20V
DWBC1,2 models, and derived analytically using the tangent method of Colomo
and Sportiello [CS16]. We intend to address this question in a future
publication.

## References

  * [And80] George E. Andrews, _Macdonald’s conjecture and descending plane partitions_ , Combinatorics, representation theory and statistical methods in groups, Lecture Notes in Pure and Appl. Math., vol. 57, Dekker, New York, 1980, pp. 91–106. MR 588196
  * [Bax89] Rodney J. Baxter, _Exactly solved models in statistical mechanics_ , Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989, Reprint of the 1982 original. MR 998375
  * [BDFZJ12] Roger E. Behrend, Philippe Di Francesco, and Paul Zinn-Justin, _On the weighted enumeration of alternating sign matrices and descending plane partitions_ , J. Combin. Theory Ser. A 119 (2012), no. 2, 331–363. MR 2860598
  * [Bre99] David M. Bressoud, _Proofs and confirmations_ , MAA Spectrum, Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 1999, The story of the alternating sign matrix conjecture. MR 1718370
  * [CS11] Luigi Cantini and Andrea Sportiello, _Proof of the Razumov-Stroganov conjecture_ , J. Combin. Theory Ser. A 118 (2011), no. 5, 1549–1574, arXiv:1003.3376 [math.CO]. MR 2771600
  * [CS16] Filippo Colomo and Andrea Sportiello, _Arctic curves of the six-vertex model on generic domains: the tangent method_ , J. Stat. Phys. 164 (2016), no. 6, 1488–1523, arXiv:1605.01388 [math-ph].
  * [DDFG20] Bryan Debin, Philippe Di Francesco, and Emmanuel Guitter, _Arctic curves of the twenty-vertex model with domain wall boundaries_ , J. Stat. Phys. 179 (2020), no. 1, 33–89. MR 4082583
  * [DFG20] Philippe Di Francesco and Emmanuel Guitter, _Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings_ , The Electronic Journal of Combinatorics 27 (2020), no. 2, P2.13, 63 pages, 22+16 figures.
  * [DFL18] Philippe Di Francesco and Matthew F. Lapa, _Arctic curves in path models from the tangent method_ , J. Phys. A: Math. Theor. 51 (2018), 155202, arXiv:1711.03182 [math-ph].
  * [GV85] Ira Gessel and Gérard Viennot, _Binomial determinants, paths, and hook length formulae_ , Adv. Math. 58 (1985), no. 3, 300–321.
  * [ICK92] Anatoli G. Izergin, David A. Coker, and Vladimir E. Korepin, _Determinant formula for the six-vertex model_ , J. Phys. A 25 (1992), no. 16, 4315–4334. MR 1181591
  * [Kel74] Stewart B. Kelland, _Twenty-vertex model on a triangular lattice_ , Australian Journal of Physics 27 (1974), 813–829.
  * [Kra06] Christian Krattenthaler, _Descending plane partitions and rhombus tilings of a hexagon with a triangular hole_ , European J. Combin. 27 (2006), no. 7, 1138–1146, arXiv:math/0310188 [math.CO]. MR 2259946
  * [Kup96] Greg Kuperberg, _Another proof of the alternative-sign matrix conjecture_ , no. 3, 139–150, arXiv:math/9712207 [math.CO].
  * [Kup02] by same author, _Symmetry classes of alternating-sign matrices under one roof_ , Ann. Math. (2002), 835–866, arXiv:math/0008184 [math.CO].
  * [Lie67] Elliott H. Lieb, _Exact solution of the problem of the entropy of two-dimensional ice_ , Phys. Rev. Lett. 18 (1967), 692–694.
  * [Lin73] Bernt Lindström, _On the vector representations of induced matroids_ , Bull. London Math. Soc. 5 (1973), no. 1, 85–90.
  * [Miz03] Hiroshi Mizukawa, _Factorization of Schur functions of various types_ , J. Algebra 269 (2003), no. 1, 215–226. MR 2015307
  * [MNdGB04] S. Mitra, B. Nienhuis, J. de Gier, and M. T. Batchelor, _Exact expressions for correlations in the ground state of the dense $\rm O(1)$ loop model_, J. Stat. Mech. Theory Exp. (2004), no. 9, 010, 24. MR 2114224
  * [MRR86] William H. Mills, David P. Robbins, and Howard Rumsey, Jr., _Self-complementary totally symmetric plane partitions_ , J. Combin. Theory Ser. A 42 (1986), no. 2, 277–292. MR 847558
  * [Pac97] Lior Pachter, _Combinatorial approaches and conjectures for 2-divisibility problems concerning domino tilings of polyominoes_ , Electron. J. Combin. 4 (1997), no. 1, R29.
  * [RS04a] A. V. Razumov and Yu. G. Stroganov, _Combinatorial nature of the ground-state vector of the $\rm O(1)$ loop model_, Teoret. Mat. Fiz. 138 (2004), no. 3, 395–400. MR 2077318
  * [RS04b] Aleksandr Vital’evich Razumov and Yu G. Stroganov, _Refined enumerations of some symmetry classes of alternating-sign matrices_ , Theor. Math. Phys. 141 (2004), no. 3, 1609–1630, arXiv:math-ph/0312071.
  * [Skl88] E. K. Sklyanin, _Boundary conditions for integrable quantum systems_ , J. Phys. A 21 (1988), no. 10, 2375–2389. MR 953215
  * [Str06] Yu. G. Stroganov, _The Izergin-Korepin determinant at a cube root of unity_ , Teoret. Mat. Fiz. 146 (2006), no. 1, 65–76. MR 2243403
  * [Tsu98] Osamu Tsuchiya, _Determinant formula for the six-vertex model with reflecting end_ , J. Math. Phys. 39 (1998), no. 11, 5946–5951. MR 1653120
  * [Zei96] Doron Zeilberger, _Proof of the alternating sign matrix conjecture_ , Electron. J. Combin. 3 (1996), no. 2, R13, arXiv:math/9407211 [math.CO].