Combinatorics of poly-Bernoulli numbers

2015/10/20 by Bényi, Beáta, Hajnal, Peter
#05A15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.05765

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 0-1 matrices of size n× 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

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