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
Categories: math.CO