Brewbaker, Chad A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues. (English) Zbl 1165.11022 Integers 8, No. 1, Article A02, 9 p. (2008). 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 \(B_x^{(-n)}+B_y^{(-n)}=B_z^{(-n)}.\) Cited in 34 Documents MSC: 11B68 Bernoulli and Euler numbers and polynomials Keywords:number of \((0,1)\)-matrices; poly-Bernoulli number; analogue of Fermat’s Little Theorem; analogue to Fermat’s Last Theorem PDFBibTeX XMLCite \textit{C. Brewbaker}, Integers 8, No. 1, Article A02, 9 p. (2008; Zbl 1165.11022) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: a(n) = Sum_{k=1..n} ((k-1)!)^2*Stirling2(n,k)^2. Array read by antidiagonals: poly-Bernoulli numbers B(-k,n).