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arXiv · 1103.4884

Poly-Bernoulli numbers and lonesum matrices

Hyun Kwang Kim·Denis S. Krotov·Joon Yop Lee
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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. \

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Hyun Kwang Kim, Denis S. Krotov, Joon Yop Lee. 2011-03-25. Poly-Bernoulli numbers and lonesum matrices. https://doi.org/10.1016/j.laa.2012.11.027

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