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A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues.

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 . 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 ≡ 2 (mod 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 +B y = B z .