Asymptotic enumeration of lonesum matrices

TL;DR

This paper derives asymptotic formulas for counting lonesum matrices using advanced combinatorial methods, and applies these results to problems in algebraic statistics and permutation enumeration.

Contribution

It introduces new asymptotic analysis techniques for lonesum matrices and connects these results to algebraic statistics and permutation enumeration.

Findings

Asymptotic formulas for bivariate poly-Bernoulli numbers

Alternative proof for square lonesum matrices using Parseval's identity

Application to asymptotic ML-degree in algebraic statistics

Abstract

We provide bivariate asymptotics for the poly-Bernoulli numbers, a combinatorial array that enumerates lonesum matrices, using the methods of Analytic Combinatorics in Several Variables (ACSV). For the diagonal asymptotic (i.e., for the special case of square lonesum matrices) we present an alternative proof based on Parseval's identity. In addition, we provide an application in Algebraic Statistics on the asymptotic ML-degree of the bivariate multinomial missing data problem, and we strengthen an existing result on asymptotic enumeration of permutations having a specified excedance set.