Asymptotic Enumeration of Integer Matrices with Constant 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.