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A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues. (English) Zbl 1165.11022

Summary: We show that the number of \((0,1)\)-matrices with \(n\) rows and \(k\) columns uniquely reconstructable from their row and column sums is the poly-Bernoulli number \(B_n^{(-k)}\). Combinatorial proofs for both the sieve and closed formulas are presented. In addition, we prove an analogue of Fermat’s Little Theorem: For a positive integer \(n\) and prime number \(p\) we have \(B_n^{(-p)}\equiv 2^n \pmod p.\) Also, an analogue to Fermat’s Last Theorem is presented: For all positive integers \(\{x,y,z\}\) and \(n>1\) there exist no solution to the equation \(B_x^{(-n)}+B_y^{(-n)}=B_z^{(-n)}.\)

MSC:

11B68 Bernoulli and Euler numbers and polynomials
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