Computation in the Enumeration of Integer Matrices

The enumeration of integer-matrices has been the subject of considerable study and it is unlikely that a simple formula exists. The number in question can be related in various ways to the representation theory of the symmetric group or of the complex general linear group, but this does not make their computation any easier. We will discuss the following three problems: (1) the number of m×n matrices over {0, 1} with each row summing to s and each column summing to t, (2) the number of nonnegative integer matrices of size m×n with each row sum equal to s and each column sum equal to t, (3) 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. We will present many conjectures based on our computation.