Asymptotic enumeration of 0–1 matrices with equal row sums and equal column sums

Let s, t, m, n be positive integers such that sm=tn. Define N(s,t;m,n) to be the number of m×n matrices with entries from {0,1}, such that each row sum is s and each column sum is t. Equivalently, N(s,t;m,n) is the number of labelled semiregular bipartite graphs, where one colour class comprises m vertices of degree s and the other comprises n vertices of degree t.A sequence of earlier papers investigated the asymptotic behaviour of N(s,t;m,n) when m,n→∞ with s and t comparatively small. The best result so far, due to McKay (1984), required s,t=o((sm)1/4). In this paper, the analysis is improved to require only the weaker condition st=o(m1/2n1/2).