带盒平面剖分的对角和

The permanent and diagonal products on the set of nonnegative matrices with bounded rank

We formulate conjectures regarding the maximum value and maximizing matrices of the permanent and of diagonal products on the set of stochastic matrices with bounded rank. We formulate equivalent conjectures on upper bounds for these functions for nonnegative matrices based on their rank, row sums and column sums. In particular we conjecture that the permanent of a singular nonnegative matrix is bounded by 1/2 times the minimum of the product of its row sums and the product of its column sums, and that the product of the elements of any diagonal of a singular nonnegative matrix is bounded by 1…

Diagonal Sums of Doubly Stochastic Matrices

Let $Ω_n$ denote the class of $n \times n$ doubly stochastic matrices (each such matrix is entrywise nonnegative and every row and column sum is 1). We study the diagonals of matrices in $Ω_n$. The main question is: which $A \in Ω_n$ are such that the diagonals in $A$ that avoid the zeros of $A$ all have the same sum of their entries. We give a characterization of such matrices, and establish several classes of patterns of such matrices.

Heisenberg XX chain, non-homogeneously parameterised generating exponential, and diagonally restricted plane partitions

The mean values of non-homogeneously parameterized generating exponential are obtained and investigated for the periodic Heisenberg XX model. The norm-trace generating function of boxed plane partitions with fixed volume of their diagonal parts is obtained as N-particles average of the generating exponential. The generating function of self-avoiding walks of random turns vicious walkers is obtained in terms of the circulant matrices that leads to generalizations of the Ramus's identity. Under various specifications of the generating exponential, the N-particles averages arise for a set of inco…

Asymptotic enumeration of symmetric integer matrices with uniform row sums

We investigate the number of symmetric matrices of non-negative integers with zero diagonal such that each row sum is the same. Equivalently, these are zero diagonal symmetric contingency tables with uniform margins, or loop-free regular multigraphs. We determine the asymptotic value of this number as the size of the matrix tends to infinity, provided the row sum is large enough. We conjecture that our answer is valid for all row sums.

正则对角矩阵下的集合划分、表与子空间轮廓

Set partitions, tableaux, and subspace profiles under regular diagonal matrices

本研究引入一族以整数划分为索引的一元多项式，其在不同取值下可对应集合划分、标准表等组合对象计数，还可用于表示第二类q-Stirling数，部分情形下与已知组合多项式一致。

改进 Blecher 和 Knopfmacher 的整数分拆不动点

Refining Blecher and Knopfmacher's Integer Partition Fixed Points

本文按固定值细化整数分拆不动点，分析所得整数三角形，证明对角、行及反对角和恒等式与四项递推，并给出组合解释。