Asymptotic enumeration of integer matrices with large equal row and column sums

Let s, t, m, n be positive integers such that sm = tn. Let M(m, s; n, t) be the number of m×n matrices over {0, 1, 2, …} with each row summing to s and each column summing to t. Equivalently, M(m, s; n, t) counts 2-way contingency tables of order m×n such that the row marginal sums are all s and the column marginal sums are all t. A third equivalent description is that M(m, s; n, t) is the number of semiregular labelled bipartite multigraphs with m vertices of degree s and n vertices of degree t. When m = n and s = t such matrices are also referred to as n×n magic or semimagic squares with line sums equal to t. We prove a precise asymptotic formula for M(m, s; n, t) which is valid over a range of (m, s; n, t) in which m, n→∞ while remaining approximately equal and the average entry is not too small. This range includes the case where m/n, n/m, s/n and t/m are bounded from below.