Some (0,1)-Matrices Enumerative Problems

摘要

Let f(m,n,s,t) be the number of (0,1) - matrices of size m × n such that each row has exactly s ones and each column has exactly t ones (sm = nt). The determination of f(m,n,s,t) is an unsolved problem, except for very small s, t. Let f_s(n) be the number of (0,1) - matrices of size n×n such that each row has exactly s 1's and each column has exactly s 1's and with the restriction that no 1 stands on the main diagonal. The determination of f_s(n) is also an unsolved problem. We will present some formulas for f(m,n,s,t) and f_s(n).