Asymptotic enumeration of sparse 0-1 matrices with irregular row and column sums

Let s = (s₁,..., s_m) and t = (t₁,..., t_n) be vectors of non-negative integer-valued functions with equal sum S = ∑_i=1^m s_i = ∑_j=1ⁿ t_j. Let N(s, t) be the number of m × n matrices with entries from {0, 1} such that the ith row has row sum s_i and the jth column has column sum t_j. Equivalently, N(s, t) is the number of labelled bipartite graphs with degrees of the vertices in one side of the bipartition given by s and the degrees of the vertices in the other side given by t. We give an asymptotic formula for N (s, t) which holds when S → ∞ with 1 ≤ st = o(S^2/3), where s = max_i s_i and t = max_j t_j. This extends a result of McKay and Wang [Linear Algebra Appl. 373 (2003) 273-288] for the semiregular case (when s_i = s for 1 ≤ i ≤ m and t_j = t for 1 ≤ j ≤ n). The previously strongest result for the non-semiregular case required 1 ≤ max{s, t} = o(S^1/4), due to McKay [Enumeration and Design, Academic Press, Canada, 1984, pp. 225-238].