Combinatorics of poly-Bernoulli numbers

TL;DR

This paper surveys various combinatorial interpretations of poly-Bernoulli numbers, introduces new interpretations, and explains Kaneko's recursive formula through combinatorial insights.

Contribution

It provides a comprehensive survey of existing combinatorial interpretations of poly-Bernoulli numbers and introduces new interpretations that clarify their properties.

Findings

New combinatorial interpretations of poly-Bernoulli numbers

A transparent explanation of Kaneko's recursive formula

Connections between poly-Bernoulli numbers and lonesum matrices

Abstract

The ${\mathbb B}_n^{(k)}$ poly-Bernoulli numbers --- a natural generalization of classical Bernoulli numbers ($B_n={\mathbb B}_n^{(1)}$) --- were introduced by Kaneko in 1997. When the parameter $k$ is negative then ${\mathbb B}_n^{(k)}$ is a nonnegative number. Brewbaker was the first to give combinatorial interpretation of these numbers. He proved that ${\mathbb B}_n^{(-k)}$ counts the so called lonesum $0\text{-}1$ matrices of size $n\times k$. Several other interpretations were pointed out. We survey these and give new ones. Our new interpretation, for example, gives a transparent, combinatorial explanation of Kaneko's recursive formula for poly-Bernoulli numbers