Jan de Gier
The appearance of numbers enumerating alternating sign matrices in stationary states of certain stochastic processes is reviewed. New conjectures concerning nest distribution functions are presented as well as a bijection between certain classes of alternating sign matrices and lozenge tilings of hexagons with cut off corners.
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We prove the equality of doubly refined enumerations of Alternating Sign Matrices and of Totally Symmetric Self-Complementary Plane Partitions using integral formulae originating from certain solutions of the quantum Knizhnik--Zamolodchikov equation.
Zeilberger proved the Refined Alternating Sign Matrix Theorem, which gives a product formula, first conjectured by Mills, Robbins and Rumsey, for the number of alternating sign matrices with given top row. Stroganov proved an explicit formula for the number of alternating sign matrices with given top and bottom rows. Fischer and Romik considered a different kind of "doubly-refined enumeration" where one counts alternating sign matrices with given top two rows, and obtained partial results on this enumeration. In this paper we continue the study of the doubly-refined enumeration with respect to…
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.
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…
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.
We prove refined enumeration results on several symmetry classes as well as related classes of alternating sign matrices with respect to classical boundary statistics, using the six-vertex model of statistical physics. More precisely, we study vertically symmetric, vertically and horizontally symmetric, vertically and horizontally perverse, off-diagonally and off-antidiagonally symmetric, vertically and off-diagonally symmetric, quarter turn symmetric as well as quasi quarter turn symmetric alternating sign matrices. Our results prove conjectures of Fischer, Duchon and Robbins.
We define a higher spin alternating sign matrix to be an integer-entry square matrix in which, for a nonnegative integer r, all complete row and column sums are r, and all partial row and column sums extending from each end of the row or column are nonnegative. Such matrices correspond to configurations of spin r/2 statistical mechanical vertex models with domain-wall boundary conditions. The case r=1 gives standard alternating sign matrices, while the case in which all matrix entries are nonnegative gives semimagic squares. We show that the higher spin alternating sign matrices of size n are …
Lecture notes for the proceedings of the workshop "Algebraic Combinatorics related to Young diagram and statistical physics", Aug. 6-10 2012, I.I.A.S., Nara, Japan.
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.
We study a one-parameter family of vector-valued polynomials associated to each simple Lie algebra. When this parameter $q$ equals -1 one recovers Joseph polynomials, whereas at $q$ cubic root of unity one obtains ground state eigenvectors of some integrable models with boundary conditions depending on the Lie algebra; in particular, we find that the sum of its entries is related to numbers of Alternating Sign Matrices and/or Plane Partitions in various symmetry classes.
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.
Limiting shape of alternating sign matrices
本文宣布均匀分布的交错符号矩阵（即等权重畴壁六顶点模型）高度函数的极限形状。
The aim of this note is to prove that fluctuations of uniformly random alternating sign matrices (equivalently, configurations of the six-vertex model with domain wall boundary conditions) near the boundary are described by the Gaussian Unitary Ensemble and the GUE-corners process.
We prove a conjecture of Cohn and Propp, which refines a conjecture of Bosley and Fidkowski about the symmetry of the set of alternating sign matrices (ASMs). We examine data arising from the representation of an ASM as a collection of paths connecting 2n vertices and show it to be invariant under the dihedral group D_{2n} rearranging those vertices, which is much bigger than the group of symmetries of the square. We also generalize conjectures of Propp and Wilson relating some of this data for different values of n.
The problem of the limit shape of large alternating sign matrices (ASMs) is addressed by studying the emptiness formation probability (EFP) in the domain-wall six-vertex model. Assuming that the limit shape arises in correspondence to the `condensation' of almost all solutions of the saddle-point equations for certain multiple integral representation for EFP, a conjectural expression for the limit shape of large ASMs is derived. The case of 3-enumerated ASMs is also considered.
We study a further refinement of the standard refined enumeration of alternating sign matrices (ASMs) according to their first two rows instead of just the first row, and more general "d-refined" enumerations of ASMs according to the first d rows. For the doubly-refined case of d=2, we derive a system of linear equations satisfied by the doubly-refined enumeration numbers A_{n,i,j} that enumerate such matrices. We give a conjectural explicit formula for A_{n,i,j} and formulate several other conjectures about the sufficiency of the linear equations to determine the A_{n,i,j}'s and about an exte…
We extend the author's formula (2011) of weighted counting of inversions on permutations to the one on alternating sign matrices. The proof is based on the sequential construction of alternating sign matrices from the unit matrix recently shown by Brualdi-Schroeder and the author (both 2017) independently.
We initiate a study of the zero-nonzero patterns of n by n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices, we find the minimum number of nonzero entries and characterize the case of equality. We also study symmetric alternating sign matrices, in particular, those with only zeros on the main diagonal. These give rise to alternating signed graphs without loops, and we determine the maximum number of edges in such graphs. We also consider n by n alternating sign mat…
An explicit expression for the numbers $A(n,r;3)$ describing the refined 3-enumeration of alternating sign matrices is given. The derivation is based on the recent results of Stroganov for the corresponding generating function. As a result, $A(n,r;3)$'s are represented as 1-fold sums which can also be written in terms of terminating ${}_4F_3$ series of argument 1/4.
Bounded Littlewood identity related to alternating sign matrices
本文建立了与交错符号矩阵相关的Littlewood恒等式推广的有界版本，并给出其两边的组合解释，旨在最终构造组合证明以理解交错符号梯形与平面分割对象的关系。
We exhibit a bijection between Dyck paths and alternating sign matrices which are determined by their antidiagonal sums.
We present multiresidue formulae for partial sums in the basis of link patterns of the polynomial solution to the level 1 U_q(\hat sl_2) quantum Knizhnik--Zamolodchikov equation at generic values of the quantum parameter q. These allow for rewriting and generalizing a recent conjecture [Di Francesco '06] connecting the above to generating polynomials for weighted Totally Symmetric Self-Complementary Plane Partitions. We reduce the corresponding conjectures to a single integral identity, yet to be proved.
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。
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。
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…
We investigate alternating sign matrices that are not permutation matrices, but have finite order in a general linear group. We classify all such examples of the form $P+T$, where $P$ is a permutation matrix and $T$ has four non-zero entries, forming a square with entries $1$ and $-1$ in each row and column. We show that the multiplicative orders of these matrices do not always coincide with those of permutation matrices of the same size. We pose the problem of identifying finite subgroups of general linear groups that are generated by alternating sign matrices.
Alternating sign matrices with a U-turn boundary (UASMs) are a recent generalization of ordinary alternating sign matrices. Here we show that variations of these matrices are in bijective correspondence with certain symplectic shifted tableaux that were recently introduced in the context of a symplectic version of Tokuyama's deformation of Weyl's denominator formula. This bijection yields a formula for the weighted enumeration of UASMs. In this connection use is made of the link between UASMs and certain square ice configuration matrices.
In alternating sign matrices the first and last nonzero entry in each row and column is specified to be +1. Such matrices always exist. We investigate a generalization by specifying independently the sign of the first and last nonzero entry in each row and column to be either a +1 or a -1. We determine necessary and sufficient conditions for such matrices to exist.
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…
We define and enumerate two new two-parameter permutation families, namely, placements of a maximum number of non-attacking rooks on $k$ chained-together $n\times n$ chessboards, in either a circular or linear configuration. The linear case with $k=1$ corresponds to standard permutations of $n$, and the circular case with $n=4$ and $k=6$ corresponds to a three-person chessboard. We give bijections of these rook placements to matrix form, one-line notation, and matchings on certain graphs. Finally, we define chained linear and circular alternating sign matrices, enumerate them for certain value…
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…
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…
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 …
The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's.
I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-function matrices, three-colorings, monotone triangles, tetrahedral order ideals, square ice, gasket-and-basket tilings and full packings of loops.
Lascoux and Schützenberger introduced a notion of key associated to any Young tableau. More recently Lascoux defined the key of an alternating sign matrix by recursively removing all -1's in such matrices. But alternating sign matrices are in bijection with monotone triangles, which form a subclass of Young tableaux. We show that in this case these two notions of keys coincide. Moreover we obtain an elegant and direct way to compute the key of any Young tableau, and discuss consequences of our result.
We prove a determinantal identity concerning Schur functions for 2-staircase diagrams lambda=(ln+l',ln,l(n-1)+l',l(n-1),...,l+l',l,l',0). When l=1 and l'=0 these functions are related to the partition function of the 6-vertex model at the combinatorial point and hence to enumerations of Alternating Sign Matrices. A consequence of our result is an identity concerning the doubly-refined enumerations of Alternating Sign Matrices.
The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn. The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's.
The p-adic valuations of a sequence of integers T(n) counting alternating sign matrices is examined for p=2 and p=3. Symmetry properties of their graphs produce a new proof of the result that characterizes the indices for which T(n) is odd.
In recent papers we have studied refined enumerations of alternating sign matrices with respect to a fixed set of top and bottom rows. The present paper is a first step towards extending these considerations to alternating sign matrices where in addition a number of left and right columns are fixed. The main result is a simple linear relation between the number of $n \times n$ alternating sign matrices where the top row as well as the left and the right column is fixed and the number of $n \times n$ alternating sign matrices where the two top rows and the bottom row is fixed. This may be seen …
We compute 2-enumerations of certain halved alternating sign matrices. In one case the enumeration equals the number of perfect matchings of a halved Aztec diamond. In the other case the enumeration equals the number of perfect matchings of a halved fortress graph. Our results prove three conjectures by Jim Propp.
We show that the poset of alternating sign matrices, with Bruhat order, is isomorphic to the poset of certain submodules of the dominant Verma module for the special linear Lie algebra $\frak{sl}_n$. The latter poset consists of the intersections of Verma submodules and can also be defined in terms of a Kazhdan-Lusztig cell.
For an integer $n\geq 1$, we display a poset $Φ_n$ whose maximal chains are in bijection with the $n\times n$ alternating sign matrices. The Hasse diagram $\widehat Φ_n$ is obtained from the $n$-cube by adding some edges. We show that the dihedral group $D_{2n}$ acts on $\widehat Φ_n$ as a group of automorphisms.
We study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices with respect to their inversion number, complementary inversion number and the position of the unique $1$ in the top row. We prove that the expansion can be expressed as a sum over totally symmetric plane partitions and we are also able to determine the coefficients. This establishes a new connection between alternating sign matrices and a class of plane partitions, thereby complementing the fact that alternating sign matrices are equinumerous with totall…
An alternating sign matrix is a square matrix with entries 1, 0 and -1 such that the sum of the entries in each row and each column is equal to 1 and the nonzero entries alternate in sign along each row and each column. To some of the symmetry classes of alternating sign matrices and their variations, G. Kuperberg associate square ice models with appropriate boundary conditions, and give determinanat and Pfaffian formulae for the partition functions. In this paper, we utilize several determinant and Pfaffian identities to evaluate Kuperberg's determinants and Pfaffians, and express the round p…
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.…
We settle a question of Bressoud concerning the existence of an explicit bijection from a class of oriented square-ice graphs to a class of tournaments. We give an algorithm constructing such a bijection.
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…
Analysis of moments and cumulants in alternating sign matrices
本文通过伯努利多项式、生成函数和可积核方法，研究了交替符号矩阵中矩和累积量的精确公式及计算方法。
Groups of singular alternating sign matrices
本文研究由奇异交错符号矩阵构成的乘法群，给出构造方法，证明任意有限群均可如此表示，并探讨矩阵大小、秩与乘法阶的关系。
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…
Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. In this paper, we prove the connectivity of alternating sign triangles, which is analogous to the connectivity of alternating sign matrices.
We introduce a new directed graph structure into the set of alternating sign matrices. This includes Bruhat graph (Bruhat order) of the symmetric groups as a subgraph (subposet). Drake-Gerrish-Skandera (2004, 2006) gave characterizations of Bruhat order in terms of total nonnegativity (TNN) and subtraction-free Laurent (SFL) expressions for permutation monomials. With our directed graph, we extend their idea in two ways: first, from permutations to alternating sign matrices; second, $q$-analogs (which we name $q$TNN and $q$SFL properties). %In our discussion, essential sets, introduced by Fult…
Enumeration of pattern-avoiding alternating sign matrices: An asymptotic dichotomy
本文对经典避免单个置换模式的交错符号矩阵的计数渐近行为给出完整分类，证明十一个特殊模式有指数上界，其余有超指数下界，并精确枚举同时避免2143和3412且含3个负一的矩阵。
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 $…
We define a bijection that transforms an alternating sign matrix A with one -1 into a pair (N,E) where N is a (so called) ``neutral'' alternating sign matrix (with one -1) and E is an integer. The bijection preserves the classical parameters of Mills, Robbins and Rumsey as well as three new parameters (including E). It translates vertical reflection of A into vertical reflection of N. A hidden symmetry allows the interchange of E with one of the remaining two new parameters. A second bijection transforms (N,E) into a configuration of lattice paths called ``mixed configuration''.
In the early 1980s, Mills, Robbins and Rumsey conjectured, and in 1996 Zeilberger proved a simple product formula for the number of $n \times n$ alternating sign matrices with a 1 at the top of the $i$-th column. We give an alternative proof of this formula using our operator formula for the number of monotone triangles with prescribed bottom row. In addition, we provide the enumeration of certain 0-1-(-1) matrices generalizing alternating sign matrices.
An alternating sign matrix is a square matrix satisfying (i) all entries are equal to 1, -1 or 0; (ii) every row and column has sum 1; (iii) in every row and column the non-zero entries alternate in sign. The 8-element group of symmetries of the square acts in an obvious way on square matrices. For any subgroup of the group of symmetries of the square we may consider the subset of matrices invariant under elements of this subgroup. There are 8 conjugacy classes of these subgroups giving rise to 8 symmetry classes of matrices. R. P. Stanley suggested the study of those alternating sign matrices…
Fischer provided a new type of binomial determinant for the number of alternating sign matrices involving the third root of unity. In this paper we prove that her formula, when replacing the third root of unity by an indeterminate $q$, is actually the $(2+q+q^{-1})$-enumeration of alternating sign matrices. By evaluating a generalisation of this determinant we are able to reprove a conjecture of Mills, Robbins and Rumsey stating that the $Q$-enumeration is a product of two polynomials in $Q$. Further we provide a closed product formula for the generalised determinant in the 0-,1- 2- and 3-enum…
We consider the groundstate wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at $Δ=-{1/2}$, along with the groundstate wavefunction of the corresponding O($n$) loop model at $n=1$. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating sign matrices and plane partitions in certain symmetry classes.
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