Related papers: Another proof of the alternating sign matrix conje…

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…

We show that, for every linear ordering of $[2]^n$, there is a large subcube on which the ordering is lexicographic. We use this to deduce that every long sequence contains a long monotone subsequence supported on an affine cube. More…

We prove a conjecture by D. Zeilberger on the determinant of a certain matrix and relate it to a problem of non-existence of 1-cycles in this note.

Let $G$ be a graph on $n$ vertices, labeled $v_1,\ldots,v_n$ and $\pi$ be a permutation on $[n]:=\{1,2,\cdots, n\}$. Suppose that each pebble $p_i$ is placed at vertex $v_{\pi(i)}$ and has destination $v_i$. During each step, a disjoint set…

We provide a generalization of the Littlewood identity, both sides of which are related to alternating sign matrices. The classical Littlewood identity establishes a nice product formula for the sum of all Schur polynomials. Compared to the…

For a system with three identical nucleons in a single-$j$ shell, the states can be written as the angular momentum coupling of a nucleon pair and the odd nucleon. The overlaps between these non-orthonormal states form a matrix which…

We introduce a new method for studying murmurations, based on random matrix theory. With this method, we exhibit murmurations or similar phenomena: assuming ratios conjectures, for elliptic curves ordered by height, quadratic twists of a…

For a Jacobi matrix J on l^2(Z_+) with Ju(n)=a_{n-1} u(n-1) + b_n u(n) + a_n u(n+1), we prove that \sum_{|E|>2} (E^2 -4)^{1/2} \leq \sum_n |b_n| + 4\sum_n |a_n -1|. We also prove bounds on higher moments and some related results in higher…

In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determinant $$\left|(i^2+dj^2)\left(\frac{i^2+dj^2}n\right)\right|_{0\le i,j\le (n-1)/2},$$ where $d$ is any…

We investigate the possible regular solutions of the boundary Yang-Baxter equation for the vertex models associated to the D_{n+1}^(2) affine Lie algebra. We have classified them in terms of three types of K-matrices. The first one have n+2…

We investigate multiplicative groups consisting entirely of singular alternating sign matrices (ASMs), and present several constructions of such groups. It is shown that every finite group is isomorphic to a group of singular ASMs, with a…

The McCarty Conjecture states that any McCarty Matrix (an $n\times n$ matrix $A$ with positive integer entries and each of the $2n$ row and column sums equal to $n$), can be additively decomposed into two other matrices, $B$ and $C$, such…

In 1961, P. Erd\H{o}s, A. Ginzburg, and A. Ziv proved a remarkable theorem stating that each set of $2n-1$ integers contains a subset of size $n$, the sum of whose elements is divisible by $n$. We will prove a similar result for pairs of…

We prove some results concerning Alcuin number of graphs. First, we classify graphs which have unique minimum vertex cover. Then we present two necessary conditions for a graph to be of class two and show why one of them (condition on…

For any three $\,n\times n\,$ matrices $\,A,B,X\,$ over a commutative ring $\,S$, we prove that $\,{\rm det}\,(A+B-AXB)={\rm det}\,(A+B-BXA) \in S$. This apparently new formula may be regarded as a ``ternary generalization'' of Sylvester's…

Guibert and Linusson introduced the family of doubly alternating Baxter permutations, i.e. Baxter permutations $\sigma \in S_n$, such that $\sigma$ and $\sigma^{-1}$ are alternating. They proved that the number of such permutations in…

Let $(P,\leq)$ be a finite poset (partially ordered set), where $P$ has cardinality $n$. Consider linear extensions of $P$ as permutations $x_1x_2\cdots x_n$ in one-line notation. For distinct elements $x,y\in P$, we define…

In this work, we study the discrete observables $$E_k = \sum_{i,j=1}^n (i-j)^k A_{i,j}$$ associated with $n\times n$ alternating sign matrices $A = (A_{i,j})$. This work develops exact formulas for expectations using Bernoulli polynomials,…

We prove that for any zero {\alpha} of the Alexander polynomial of a two-bridge knot, -3 < Re({\alpha}) < 6. Furthermore, for a large class of two-bridge knots we prove -1<Re({\alpha}).

We explore the enumeration of some natural classes of graded posets, including all graded posets, (2+2)- and (3+1)-avoiding graded posets, (2+2)-avoiding graded posets, and (3+1)-avoiding graded posets. We obtain enumerative and structural…