In mathematics, poly-Bernoulli numbers, denoted as , were defined by M. Kaneko as where Li is the polylogarithm. The are the usual Bernoulli numbers. Moreover, the Generalization of Poly-Bernoulli numbers with a,b,c parameters defined as follows where Li is the polylogarithm. Kaneko also gave two combinatorial formulas: where is the number of ways to partition a size set into non-empty subsets (the Stirling number of the second kind). A combinatorial interpretation is that the poly-Bernoulli numbers of negative index enumerate the set of by (0,1)-matrices uniquely reconstructible from their row and column sums. Also it is the number of open tours by a biased rook on a board (see A329718 for definition). The Poly-Bernoulli number satisfies the following asymptotic: For a positive integer n and a prime number p, the poly-Bernoulli numbers satisfy which can be seen as an analog of Fermat's little theorem. Further, the equation has no solution for integers x, y, z, n > 2; an analog of Fermat's Last Theorem. Moreover, there is an analogue of Poly-Bernoulli numbers (like Bernoulli numbers and Euler numbers) which is known as Poly-Euler numbers. See also Bernoulli numbers Stirling numbers Gregory coefficients Bernoulli polynomials Bernoulli polynomials of the second kind Stirling polynomials References . . . . Integer sequences Enumerative combinatorics