On the refined 3-enumeration of alternating sign matrices

The Rotor Model with spectral parameters and enumerations of Alternating Sign Matrices

In this paper we study the Rotor Model of Martins and Nienhuis. After introducing spectral parameters, a combined use of integrability, polynomiality of the ground state wave function and a mapping into the fully-packed O(1)-model allows us to determine the sum rule and a family of maximally nested components for different boundary conditions. We see in this way the appearance of 3-enumerations of Alternating Sign Matrices.

Enumerations of half-turn symmetric alternating-sign matrices of odd order

It was shown by Kuperberg that the partition function of the square-ice model related to half-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with half-turn symmetric alternating-sign matrices of odd order. The partition function of the model is expressed via the above mentioned factors. The contributions to the partition function of the states corresponding to the alternating-sign matrices having 1 or -1 as the central entry are found and the related enumerations are obtained.

New enumeration formulas for alternating sign matrices and square ice partition functions

The refined enumeration of alternating sign matrices (ASMs) of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study, starting with the Refined Alternating Sign Matrix Conjecture of Mills-Robbins-Rumsey, its proof by Zeilberger, and more recent work on doubly-refined and triply-refined enumeration by several authors. In this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the "top-left-bottom" (triply-refined) and "top-left-bottom-right" (quadruply-refined) enumerations.…

Enumeration of quarter-turn symmetric alternating-sign matrices of odd order

It was shown by Kuperberg that the partition function of the square-ice model related to the quarter-turn symmetric alternating-sign matrices of even order is the product of two similar factors. We propose a square-ice model whose states are in bijection with the quarter-turn symmetric alternating-sign matrices of odd order, and show that the partition function of this model can be also written in a similar way. This allows to prove, in particular, the conjectures by Robbins related to the enumeration of the quarter-turn symmetric alternating-sign matrices.

Enumeration of alternating sign triangles using a constant term approach

Alternating sign triangles (ASTs) have recently been introduced by Ayyer, Behrend and the author, and it was proven that there is the same number of ASTs with n rows as there is of nxn alternating sign matrices (ASMs). We prove a conjecture by Behrend on a refined enumeration of ASTs with respect to a statistic that is shown to have the same distribution as the column of the unique 1 in the top row of an ASM. The proof of the conjecture is based on a certain multivariate generating function of ASTs that takes the positions of the columns with sum 1 (1-columns) into account. We also prove a cur…

具有交替边界的通用平面超图的计数

Enumeration of general planar hypermaps with an alternating boundary

本文扩展了之前工作中关于具有交替边界的平面超图的枚举研究，通过经典的一对多对应关系，提出了一种新的策略来获得一般情况下的代数方程，包括装饰有伊辛模型的图，并通过同时消除两个催化变量的主要步骤，展示了伊辛四边形化情况下的显式有理参数化，从而证明了一些显著性质在一般情况下不再成立。

Multiply-refined enumeration of alternating sign matrices

Four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices (ASMs). Specifically, these statistics are the positions of the 1's in the first and last rows and columns of an ASM, and the numbers of generalized inversions and -1's in an ASM. Previously-known and related results for the exact enumeration of ASMs with prescribed values of some of these statistics are discussed in detail. A quadratic relation which recursively determines the generating function associated with all six statistics is then obtained. This relation also leads to various …

Square ice, alternating sign matrices and classical orthogonal polynomials

The six-vertex model with Domain Wall Boundary Conditions, or square ice, is considered for particular values of its parameters, corresponding to 1-, 2-, and 3-enumerations of Alternating Sign Matrices (ASMs). Using Hankel determinant representations for the partition function and the boundary correlator of homogeneous square ice, it is shown how the ordinary and refined enumerations can be derived in a very simple and straightforward way. The derivation is based on the standard relationship between Hankel determinants and orthogonal polynomials. For the particular sets of parameters correspon…

The Combinatorics of Alternating Tangles: from theory to computerized enumeration

We study the enumeration of alternating links and tangles, considered up to topological (flype) equivalences. A weight $n$ is given to each connected component, and in particular the limit $n\to 0$ yields information about (alternating) knots. Using a finite renormalization scheme for an associated matrix model, we first reduce the task to that of enumerating planar tetravalent diagrams with two types of vertices (self-intersections and tangencies), where now the subtle issue of topological equivalences has been eliminated. The number of such diagrams with $p$ vertices scales as $12^p$ for $p\…

交错符号矩阵的冻结角枚举

Frozen-corner enumeration of Alternating Sign Matrices

研究交错符号矩阵的冻结角枚举问题，给出基于特定s×s矩阵行列式的推测公式，通过数值支持该推测，并阐述其与大型交错符号矩阵极限形状、涨落及Tracy - Widom分布的关联。

Bethe roots and refined enumeration of alternating-sign matrices

The properties of the most probable ground state candidate for the XXZ spin chain with the anisotropy parameter equal to -1/2 and an odd number of sites is considered. Some linear combinations of the components of the considered state, divided by the maximal component, coincide with the elementary symmetric polynomials in the corresponding Bethe roots. It is proved that those polynomials are equal to the numbers providing the refined enumeration of the alternating-sign matrices of order M+1 divided by the total number of the alternating-sign matrices of order M, for the chain of length 2M+1.

An Izergin-Korepin-type identity for the 8VSOS model, with applications to alternating sign matrices

We obtain a new expression for the partition function of the 8VSOS model with domain wall boundary conditions, which we consider to be the natural extension of the Izergin-Korepin formula for the six-vertex model. As applications, we find dynamical (in the sense of the dynamical Yang-Baxter equation) generalizations of the enumeration and 2-enumeration of alternating sign matrices. The dynamical enumeration has a nice interpretation in terms of three-colourings of the square lattice.

Constant term formulas for refined enumerations of Gog and Magog trapezoids

Gog and Magog trapezoids are certain arrays of positive integers that generalize alternating sign matrices (ASMs) and totally symmetric self-complementary plane partitions (TSSCPPs) respectively. Zeilberger used constant term formulas to prove that there is the same number of (n,k)-Gog trapezoids as there is of (n,k)-Magog trapezoids, thereby providing so far the only proof for a weak version of a conjecture by Mills, Robbins and Rumsey from 1986. About 20 years ago, Krattenthaler generalized Gog and Magog trapezoids and formulated an extension of their conjecture, and, recently, Biane and Che…

A Transfer Matrix approach to the Enumeration of Knots

We propose a new method to enumerate alternating knots using a transfer matrix approach. We apply it to count numerically various objects, including prime alternating tangles with two connected components, up to order 18--22, and comment on the large-order behavior in connection with one of the authors' conjecture.

On the weighted enumeration of alternating sign matrices and descending plane partitions

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359] that, for any n, k, m and p, the number of nxn alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m -1's and m+p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with f…

Enumeration of planar constellations with an alternating boundary

A planar hypermap with a boundary is defined as a planar map with a boundary, endowed with a proper bicoloring of the inner faces. The boundary is said alternating if the colors of the incident inner faces alternate along its contour. In this paper we consider the problem of counting planar hypermaps with an alternating boundary, according to the perimeter and to the degree distribution of innerfaces of each color. The problem is translated into a functional equation with a catalytic variable determining the corresponding generating function. In the case of constellations - hypermaps whose all…

A transfer matrix approach to the enumeration of colored links

We propose a transfer matrix algorithm for the enumeration of alternating link diagrams with external legs, giving a weight $n$ to each connected component. Considering more general tetravalent diagrams with self-intersections and tangencies allows us to treat topological (flype) equivalences. This is done by means of a finite renormalization scheme for an associated matrix model. We give results, expressed as polynomials in $n$, for the various generating functions up to order 19 (link diagrams), 15 (prime alternating tangles) and 11 (6-legged links) intersections. The limit $n\to\infty$ is s…

A doubly-refined enumeration of alternating sign matrices and descending plane partitions

It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on nxn alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The statistics for an ASM A are the number of generalized inversions in A, the number of -1's in A and the number of 0's to the left of the 1 in the first row of A, and the respective statistics for a DPP D are the number of nonspecial parts in D, the number of special parts in D and the number of n's in D. Here, the result is generalized to include a fourth st…

对角对称交替符号矩阵

Diagonally symmetric alternating sign matrices

本文研究对角对称交替符号矩阵（DSASMs）的计数，得到其数量及相关统计量生成函数的Pfaffian公式，引入与DSASM一一对应的六顶点模型变体并研究相关主题。

Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices

We prove and conjecture some new symmetric function identities, which equate the generating series of 1. Plane partitions, subject to certain restrictions and weightings, and 2. Alternating sign matrices, subject to certain symmetry properties. The left hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity, allowing them to be interpreted as generating series for plane partitions. The right hand side of each identity is a partition function of the six-vertex model, on a relevant domain. These can be interpreted as generating series for alternati…

A refined Razumov-Stroganov conjecture II

We extend a previous conjecture [cond-mat/0407477] relating the Perron-Frobenius eigenvector of the monodromy matrix of the O(1) loop model to refined numbers of alternating sign matrices. By considering the O(1) loop model on a semi-infinite cylinder with dislocations, we obtain the generating function for alternating sign matrices with prescribed positions of 1's on their top and bottom rows. This seems to indicate a deep correspondence between observables in both models.

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbin…

具有指定底行的单调梯形的数量

The number of monotone trapezoids with prescribed bottom row

本文建立了具有指定底行的单调梯形数量的算子公式，推广交替符号矩阵，发现特殊情况下被消去的隐藏算子，涉及自由边界并揭示与共变代数的关系，提出相关猜想。

Symmetry classes of alternating sign matrices in the nineteen-vertex model

The nineteen-vertex model on a periodic lattice with an anti-diagonal twist is investigated. Its inhomogeneous transfer matrix is shown to have a simple eigenvalue, with the corresponding eigenstate displaying intriguing combinatorial features. Similar results were previously found for the same model with a diagonal twist. The eigenstate for the anti-diagonal twist is explicitly constructed using the quantum separation of variables technique. A number of sum rules and special components are computed and expressed in terms of Kuperberg's determinants for partition functions of the inhomogeneous…

A refined Razumov-Stroganov conjecture

We extend the Razumov-Stroganov conjecture relating the groundstate of the O(1) spin chain to alternating sign matrices, by relating the groundstate of the monodromy matrix of the O(1) model to the so-called refined alternating sign matrices, i.e. with prescribed configuration of their first row, as well as to refined fully-packed loop configurations on a square grid, keeping track both of the loop connectivity and of the configuration of their top row. We also conjecture a direct relation between this groundstate and refined totally symmetric self-complementary plane partitions, namely, in th…

Loop model with mixed boundary conditions, qKZ equation and alternating sign matrices

The integrable loop model with mixed boundary conditions based on the 1-boundary extended Temperley--Lieb algebra with loop weight 1 is considered. The corresponding qKZ equation is introduced and its minimal degree solution described. As a result, the sum of the properly normalized components of the ground state in size L is computed and shown to be equal to the number of Horizontally and Vertically Symmetric Alternating Sign Matrices of size 2L+3. A refined counting is also considered.

3-enumerated alternating sign matrices

Let $A(n,r;3)$ be the total weight of the alternating sign matrices of order $n$ whose sole `1' of the first row is at the $r^{th}$ column and the weight of an individual matrix is $3^k$ if it has $k$ entries equal to -1. Define the sequence of the generating functions $G_n(t)=\sum_{r=1}^n A(n,r;3)t^{r-1}$. Results of two different kind are obtained. On the one hand I made the explicit expression for the even subsequence $G_{2ν}(t)$ in terms of two linear homogeneous second order recurrence in $ν$ (Theorem 1). On the other hand I brought to light the nice connection between the neighbouring fu…

Alternating sign matrices and totally symmetric plane partitions

We introduce a new family $\mathcal{A}_{n,k}$ of Schur positive symmetric functions, which are defined as sums over totally symmetric plane partitions. In the first part, we show that, for $k=1$, this family is equal to a multivariate generating function involving $n+3$ variables of objects that extend alternating sign matrices (ASMs), which have recently been introduced by the authors. This establishes a new connection between ASMs and a class of plane partitions, thereby complementing the fact that ASMs are equinumerous with totally symmetric self-complementary plane partitions as well as wi…

交替符号五边形与 Magog 五边形

Alternating Sign Pentagons and Magog Pentagons

本文通过统计交替符号三角形左右全 0 列定义交替符号五边形，证明其与特定 Magog 五边形在统计量 $\ ho$ 下等数，并给出 Pfaffian 形式的生成函数。

$\mathfrak{b}$-Hurwitz数从精炼的拓扑递归

$\mathfrak{b}$-Hurwitz numbers from refined topological recursion

本文基于精炼拓扑递归计算$\mathfrak{b}$-Hurwitz数，证明其在有理谱曲线上的解析延拓，并涵盖$\mathfrak{b}$-单调Hurwitz数及非定向曲面地图计数，应用于Gaussian等$β$-ensembles的相关子集计算。

The nineteen-vertex model and alternating sign matrices

It is shown that the transfer matrix of the inhomogeneous nineteen-vertex model with certain diagonal twisted boundary conditions possesses a simple eigenvalue. This is achieved through the identification of a simple and completely explicit solution of its Bethe equations. The corresponding eigenvector is computed by means of the algebraic Bethe ansatz, and both a simple component and its square norm are expressed in terms of the Izergin-Korepin determinant. In the homogeneous limit, the vector coincides with a supersymmetry singlet of the twisted spin-one XXZ chain. It is shown that in a natu…

各向异性参数$Δ=-1/2$的开放XXZ链与全对称交替符号矩阵

The open XXZ chain at $Δ=-1/2$ and totally-symmetric alternating sign matrices

本研究针对各向异性参数$Δ=-1/2$、含$x$依赖对角边界场的有限长开放XXZ自旋链，揭示其基态矢量分量为整数系数$x$多项式，且分量和对应全对称交替符号矩阵加权计数的生成函数。

与交替符号矩阵相关的新双射证明

New bijective proofs pertaining to alternating sign matrices

本文针对 ASM-DPP 双射问题，利用相容性概念构造交替符号矩阵与移位 Gelfand-Tsetlin 模式间更自然的带符号双射，并为含 n+3 个统计量的 ASM 推广的精细化枚举提供双射证明。

The relation between alternating sign matrices and descending plane partitions: $n+3$ pairs of equivalent statistics

There is the same number of $n \times n$ alternating sign matrices (ASMs) as there is of descending plane partitions (DPPs) with parts no greater than $n$, but finding an explicit bijection is an open problem for about $40$ years now. So far, quadruples of statistics on ASMs and on DPPs that have the same joint distribution have been identified. We introduce extensions of ASMs and of DPPs along with $n+3$ statistics on each extension, and show that the two families of statistics have the same joint distribution. The ASM-DPP equinumerosity is obtained as an easy consequence by considering the $…

Izergin-Korepin determinant reloaded

We consider the Izergin-Korepin determinant [1] together with another determinant which was invented by Kuperberg [2]. He used these determinants to prove a formula for the total number of half-turn symmetric alternating sign matrices of even order conjectured by Robbins [3]. By developing further the method that was described in our previous paper [4], we obtain a closed nonlinear recurrence system for these determinants. It can be used in various ways. For example, in this paper, we obtain formula (29) for the refined enumeration of half-turn symmetric alternating sign matrices of even order…

Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order

For each $α\in \{0,1,-1 \}$, we count diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal number of $α$'s along the diagonal and the antidiagonal, as well as DASASMs of fixed odd order with a minimal number of $0$'s along the diagonal and the antidiagonal. In these enumerations, we encounter product formulas that have previously appeared in plane partition or alternating sign matrix counting, namely for the number of all alternating sign matrices, the number of cyclically symmetric plane partitions in a given box, and the number of vert…

交错符号矩阵与下降平面分拆之间的局部双射及 Striker--Fulmek 型 $q$-统计量

A local bijection between alternating sign matrices and descending plane partitions and a Striker--Fulmek-type $q$-statistic

本文构造了 Behrend--Di Francesco--Zinn-Justin 细化纤维中交错符号矩阵与下降平面分拆之间的局部双射，通过移动边界线实现，并引入了一个保持统计量的 Striker--Fulmek 型 $q$-统计量。