Related papers: 3-enumerated alternating sign matrices

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…

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…

Since the alternating sign matrix conjecture, proposed by Mills, Robbins, and Rumsey in 1982, was proved by Zeilberger and Kuperberg, several refined enumerations have been considered. In particular, Behrend et al. obtained a quadruply…

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…

Let $m$, $k_1$, and $k_2$ be three integers with $m\ge 2$. For any set $A\subseteq \mathbb{Z}_m$ and $n\in \mathbb{Z}_m$, let $\hat{r}_{k_1,k_2}(A,n)$ denote the number of solutions of the equation $n=k_1a_1+k_2a_2$ with $a_1,a_2\in A$. In…

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…

Let $f_n(x_1, x_2, \ldots, x_n)$ denote the algebraic normal form (polynomial form) of a rotation symmetric Boolean function of degree $d$ in $n \geq d$ variables and let $wt(f_n)$ denote the Hamming weight of this function. Let $(1, a_2,…

Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\odot S=\{w_1s_1+...+w_ns_n:\;w_i…

The binary operation of usual addition is associative in all common matrices over R. However, here we define a binary operation of addition in matrices over Zn which present the concept of nonassociativity. These structures form Matrix…

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…

Arrowed Gelfand-Tsetlin patterns have recently been introduced to study alternating sign matrices. In this paper, we show that a $(-1)$-enumeration of arrowed Gelfand-Tsetlin patterns can be expressed by a simple product formula. The…

We principally present reductions of certain generalized hypergeometric functions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most of these results have been known for some time, but one of the methods, wherein we…

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…

Consider the following noncommutative arithmetic-geometric mean inequality: given positive-semidefinite matrices $\mathbf{A}_1, \dots, \mathbf{A}_n$, the following holds for each integer $m \leq n$: $$ \frac{1}{n^m}\sum_{j_1, j_2, \dots,…

This paper highlights three known identities, each of which involves sums over alternating sign matrices. While proofs of all three are known, the only known derivations are as corollaries of difficult results. The simplicity and natural…

Monotone triangles are plane integer arrays of triangular shape with certain monotonicity conditions along rows and diagonals. Their significance is mainly due to the fact that they correspond to $n \times n$ alternating sign matrices when…

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…

We derive a simple functional equation with two catalytic variables characterising the generating function of 3-stack-sortable permutations. Using this functional equation, we extend the 174-term series to 1000 terms. From this series, we…

In this paper we show that if for an integer matrix A the universal Gr\"obner basis of the associated toric ideal \Ideal_A coincides with the Graver basis of A, then the Gr\"obner complexity u(A) and the Graver complexity g(A) of its higher…

We initiate the study of enumerating linear subspaces of alternating matrices over finite fields with explicit coordinates. We postulate that this study can be viewed as a linear algebraic analogue of the classical topic of enumerating…