Related papers: Proof of the Alternating Sign Matrix Conjecture

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of…

We prove bijectively that the total number of cycles of all even permutations of $[n]=\{1,2,...,n\}$ and the total number of cycles of all odd permutations of $[n]$ differ by $(-1)^n(n-2)!$, which was stated as an open problem by Mikl\'{o}s…

If $M$ is a set of nonsingular $k\times k$ matrices then for many pairs of matrices, $A,B\in M,$ the sum is nonsingular, $\det(A+B)\neq 0.$ We prove a more general statement on nonsingular sums with an application.

The number of non-negative integer matrices with given row and column sums appears in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations of various kinds. Here we…

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…

A combinatorial rectangle may be viewed as a matrix whose entries are all +-1. The discrepancy of an m by n matrix is the maximum among the absolute values of its m row sums and n column sums. In this paper, we investigate combinatorial…

Let $n>1$ and $k>0$ be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix with $m$ rows is called reducible if we can delete $j$ rows, $0<j<m$, so that the remaining matrix is level. We ask if…

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…

Let $A$ be a real matrix. The term rank of $A$ is the smallest number $t$ of lines (that is, rows or columns) needed to cover all the nonzero entries of $A$. We prove a conjecture of Li et al. stating that, if the rank of $A$ exceeds $t-3$,…

The sign patterns of inverse doubly-nonnegative matrices are examined. A necessary and sufficient condition is developed for a sign matrix to correspond to an inverse doubly-nonnegative matrix. In addition, for a doubly-nonnegative matrix…

A {\em signed magic rectangle} $SMR(m,n;k, s)$ is an $m \times n$ array with entries from $X$, where $X=\{0,\pm1,\pm2,\ldots, $ $\pm (mk-1)/2\}$ if $mk$ is odd and $X = \{\pm1,\pm2,\ldots,\pm mk/2\}$ if $mk$ is even, such that precisely $k$…

The number of ``carries'' when $n$ random integers are added forms a Markov chain [23]. We show that this Markov chain has the same transition matrix as the descent process when a deck of $n$ cards is repeatedly riffle shuffled. This gives…

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 consider the notion of a signed magic array, which is an $m \times n$ rectangular array with the same number of filled cells $s$ in each row and the same number of filled cells $t$ in each column, filled with a certain set of numbers…

It is shown that a random $(0,1)$ matrix whose rows are independent random vectors of exactly $n/2$ zero components is non-singular with probability $1-O(n^{-C})$ for any $C>0$. The proof uses a non-standard inverse-type Littlewood-Offord…

Let $I_k = [(2k-1)^2, (2k+1)^2)$ for $k \geq 1$. Starting from the odd-composite matrix $(b_{ij})$ with $b_{ij} = (2i-1)(2j-1)$, introduced by the author in [1], we define for each odd integer $n$ the \emph{matrix multiplicity} $r(n)$, 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…

An n x n permutative matrix is a matrix in which every row is a permutation of the first row. In this paper the result given by Paparella in [Electron. J. Linear Algebra 31 (2016) 306-312] is extended to a more general lists of real and…

Consider a random $n\times n$ zero-one matrix with "density" $p$, sampled according to one of the following two models: either every entry is independently taken to be one with probability $p$ (the "Bernoulli" model), or each row is…

We prove upper and lower bounds on the number of pairs of commuting $n\times n$ matrices with integer entries in $[-T,T]$, as $T\to \infty$. Our work uses Fourier analysis and leads us to an analysis of exponential sums involving matrices…