Poly-Bernoulli numbers and lonesum matrices

TL;DR

This paper extends the concept of lonesum matrices to q-ary cases, computes their counts, and generalizes formulas related to poly-Bernoulli numbers, opening avenues for further research.

Contribution

It introduces q-ary lonesum matrices, computes their enumeration, and generalizes existing formulas for poly-Bernoulli numbers, expanding the theoretical framework.

Findings

Number of q-ary lonesum matrices computed

Generalized formulas for poly-Bernoulli numbers derived

Defined strong and weak q-ary lonesum matrices

Abstract

A lonesum matrix is a matrix that can be uniquely reconstructed from its row and column sums. Kaneko defined the poly-Bernoulli numbers $B_m^{(n)}$ by a generating function, and Brewbaker computed the number of binary lonesum $m\times n$-matrices and showed that this number coincides with the poly-Bernoulli number $B_m^{(-n)}$. We compute the number of $q$-ary lonesum $m\times n$-matrices, and then provide generalized Kaneko's formulas by using the generating function for the number of $q$-ary lonesum $m\times n$-matrices. In addition, we define two types of $q$-ary lonesum matrices that are composed of strong and weak lonesum matrices, and suggest further researches on lonesum matrices. \