Lonesum (0,1)-matrices and poly-Bernoulli numbers of negative index

This thesis shows that the number of (0,1)-matrices with n rows and k columns uniquely reconstructible from their row and column sums are the poly-Bernoulli numbers of negative index, B[subscript n superscript ( -k)] . Two proofs of this main theorem are presented giving a combinatorial bijection between two poly-Bernoulli formula found in the literature. Next, some connections to Fermat are proved showing that for a positive integer n and prime number p B[subscript n superscript ( -p) congruent 2 superscript n (mod p),] and that for all positive integers (x, y, z, n) greater than two there exist no solutions to the equation: B[subscript x superscript ( -n)] + B[subscript y superscript ( -n)] = B[subscript z superscript ( -n)]. In addition directed graphs with sum reconstructible adjacency matrices are surveyed, and enumerations of similar (0,1)-matrix sets are given as an appendix

The number of (0,1) - Matrices with fixed row and column sums

Let R and S be non-negative and non-increasing vectors of order m and n respectively. We consider the set A(R, S) of all m x n matrices with entries restricted to {0, 1}. We give an alternative proof of the Gale-Ryser theorem, which determines when A(R, S) is non-empty. We show conditions for R and S so that ∣A(R, S) ∣ ∈ {1, n!}. We also examine the case where ∣A(R, S) ∣ = 2 and describe the structure of those matrices. We show that for each positive integer k, there is a possible choice of R and S so that ∣A(R, S) ∣ = k. Furthermore, we explore gm,n(x; y), the generating function for the cardinality ∣A(R, S) ∣ of all possible combinations of R and S

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application