Table of minimum ranks of graphs of order at most 7 and selected optimal matrices

Asymptotic estimates for the number of contingency tables, integer flows, and volumes of transportation polytopes

We prove an asymptotic estimate for the number of mxn non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer feasible flows in a network. Similarly, we estimate the volume of the polytope of mxn non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems and hence can be computed efficiently. As a corollary, we show that if row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n) with r_1 + ... + r_m =c_1 + ... +c_n =N are sufficiently far from cons…

真圆弧图是 $e$-正的

Proper circular arc graphs are $e$-positive

本文证明了真圆弧图的色对称函数是 $e$-正的，解决了 Ellzey 猜想在 $q=1$ 时的情形，并通过颜色矩阵与表矩阵的迹给出显式公式，同时为 Stanley--Stembridge 猜想提供了新证明。

A Short Note on Asymptotic Enumeration of Contingency Tables with Non-Uniform Margins

In this short note, we compute the precise asymptotics for the number of contingency tables with non-uniform margins. More precisely, for parameter $n,δ, B,C>0$, we consider the set of matrices whose first $[n^δ]$ rows and columns have sum $[BCn]$ and the rest $n$ rows and columns have sum $[Cn]$. We compute the precise asymptotics of the cardinality of this set when $B<B_c=1+\sqrt{1+1/C}$ using the maximal entropy methods developed by Barvinok and Hartigan. The only contribution of this note is a detailed expansion of the determinant of quadratic forms in asymptotic formulas.

Decomposing labeled interval orders as pairs of permutations

We introduce ballot matrices, a signed combinatorial structure whose definition naturally follows from the generating function for labeled interval orders. A sign reversing involution on ballot matrices is defined. We show that matrices fixed under this involution are in bijection with labeled interval orders and that they decompose to a pair consisting of a permutation and an inversion table. To fully classify such pairs, results pertaining to the enumeration of permutations having a given set of ascent bottoms are given. This allows for a new formula for the number of labeled interval orders…

Hadamard matrices from base sequences: An example

There are several well-known methods that one can use to construct Hadamard matrices from base sequences BS(m,n). In view of the recent classification of base sequences BS(n+1,n) for n <= 30, it may be of interest to show on an example how prolific these methods are. For that purpose we have selected the Hadamard matrices of order 60. By using these methods and the transposition map we have constructed 1759 nonequivalent Hadamard matrices of order 60.

Low rank approximations of symmetric polynomials and asymptotic counting of contingency tables

We represent the number of mxn non-negative integer matrices (contingency tables) with prescribed row sums and column sums as the expected value of the permanent of a non-negative random matrix with exponentially distributed entries. We bound the variance of the obtained estimator, from which it follows that if the row and column sums are bounded by a constant fixed in advance, we get a polynomial time approximation scheme for counting contingency tables. We show that the complete symmetric polynomial of a fixed degree in n variables can be epsilon-approximated coefficient-wise by a sum of pow…

一个 Hadamard 矩阵构造数据库

A database of constructions of Hadamard matrices

本文在 SageMath 中构建覆盖阶≤1208的 Hadamard 与斜 Hadamard 矩阵构造数据库，修正文献错误，并更新至阶2999/999的最小指数表，还揭示其与 Riesel 数的关系。

A note on identities in plactic monoids and monoids of upper-triangular tropical matrices

This paper uses the combinatorics of Young tableaux to prove the plactic monoid of infinite rank does not satisfy a non-trivial identity, by showing that the plactic monoid of rank $n$ cannot satisfy a non-trivial identity of length less than or equal to $n$. A new identity is then proven to hold for the monoid of $n \times n$ upper-triangular tropical matrices. Finally, a straightforward embedding is exhibited of the plactic monoid of rank $3$ into the direct product of two copies of the monoid of $3\times 3$ upper-triangular tropical matrices, giving a new proof that the plactic monoid of ra…

Computation of Maximal Determinants of Binary Circulant Matrices

We describe algorithms for computing maximal determinants of binary circulant matrices of small orders. Here "binary matrix" means a matrix whose elements are drawn from $\{0,1\}$ or $\{-1,1\}$. We describe efficient parallel algorithms for the search, using Duval's algorithm for generation of necklaces and the well-known representation of the determinant of a circulant in terms of roots of unity. Tables of maximal determinants are given for orders $\le 53$. Our computations extend earlier results and disprove two plausible conjectures.

Integrity of ghosts

Sequences of traces of powers of integral matrices are characterized.The result can be used can be used in character tables of e.g. the symmetric group.

具有给定边距的大型随机矩阵

Large random matrices with given margins

本文研究了具有给定行和与列和边距的大型i.i.d.随机矩阵，提出了用独立指数倾斜矩阵近似条件矩阵的“转移原理”，并解决了多个相关猜想。

An attack on Zarankiewicz's problem through SAT solving

The Zarankiewicz function gives, for a chosen matrix and minor size, the maximum number of ones in a binary matrix not containing an all-one minor. Tables of this function for small arguments have been compiled, but errors are known in them. We both correct the errors and extend these tables in the case of square minors by expressing the problem of finding the value at a specific point as a series of Boolean satisfiability problems, exploiting permutation symmetries for a significant reduction in the work needed. When the ambient matrix is also square we also give all non-isomorphic examples o…

连续时间量子游走中的若干开放性问题精选

Selected Open Problems in Continuous-Time Quantum Walks

本文综述连续时间量子游走领域，梳理完美态传输、瞬时均匀混合和平均混合矩阵三类开放问题，以促进量子计算与代数图论交叉研究。