Related papers: Another proof of the alternating sign matrix conje…
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…
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…
Alternating sign triangles have been introduced by Ayyer, Behrend and Fischer in 2016 and it was proven that there is the same number of alternating sign triangles with $n$ rows as there is of $n\times n$ alternating sign matrices. Later on…
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…
Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2].…
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…
Vertically symmetric alternating sign matrices (VSASMs) of order $2n+1$ are known to be equinumerous with lozenge tilings of a hexagon with side lengths $2n+2$, $2n$, $2n+2$, $2n$, $2n+2$, $2n$ and a central triangular hole of size $2$ that…
In a previous article [math.CO/9712207], we derived the alternating-sign matrix (ASM) theorem from the Izergin-Korepin determinant for a partition function for square ice with domain wall boundary. Here we show that the same argument…
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…
The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $N$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free…
Alternating sign triangles were introduced by Carroll and Speyer in relation to cube recurrence, by analogy to alternating sign matrices for octahedron recurrence. Permutation triangles are the alternating sign triangles whose entries are…
Let $ R \subset \R $ be a GCD-domain. In this paper, Weinberg's conjecture on the $ n \times n $ matrix algebra $ M_{n}(R) \ (n \geq 2) $ is proved. Moreover, all the lattice orders (up to isomorphisms) on a full $ 2 \times 2 $ matrix…
We provide the first example of a sign pattern $S$ for which there exist orthogonal matrices $Q_1$ and $Q_2$ with sign pattern $S$ such that $\det Q_1=1$ and $\det Q_2=-1$. The existence of such matrices is raised by C. Waters in {"Sign…
An ordered matching of size $n$ is a graph on a linearly ordered vertex set $V$, $|V|=2n$, consisting of $n$ pairwise disjoint edges. There are three different ordered matchings of size two on $V=\{1,2,3,4\}$: an alignment…
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…
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…
We show that for fixed $d>3$ and $n$ growing to infinity there are at least $(n!)^{d-2 \pm o(1)}$ different labeled combinatorial types of $d$-polytopes with $n$ vertices. This is about the square of the previous best lower bounds. As an…
An alternating sign matrix, or ASM, is a $(0, \pm 1)$-matrix where the nonzero entries in each row and column alternate in sign. We generalize this notion to hypermatrices: an $n\times n\times n$ hypermatrix $A=[a_{ijk}]$ is an {\em…
Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We put ASMs into a larger context by studying the order ideals of subposets of a…
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…