Asymptotic enumeration of sparse nonnegative integer matrices with specified row and column sums

Let s = (s₁, ..., s_m) and t = (t₁, ..., t_n) be vectors of nonnegative integer-valued functions of m, n with equal sum S = ∑_{i = 1}^m s_i = ∑_{j = 1}ⁿ t_j. Let M (s, t) be the number of m × n matrices with nonnegative integer entries such that the ith row has row sum s_i and the jth column has column sum t_j for all i, j. Such matrices occur in many different settings, an important example being the contingency tables (also called frequency tables) important in statistics. Define s = max_i s_i and t = max_j t_j. Previous work has established the asymptotic value of M (s, t) as m, n → ∞ with s and t bounded (various authors independently, 1971-1974), and when all entries of s equal s, all entries of t equal t, and m / n, n / m, s / n ≥ c / log n for sufficiently large c [E.R. Canfield, B.D. McKay, Asymptotic enumeration of integer matrices with large equal row and column sums, submitted for publication, 2007]. In this paper we extend the sparse range to the case s t = o (S^{2 / 3}). The proof in part follows a previous asymptotic enumeration of 0-1 matrices under the same conditions [C. Greenhill, B.D. McKay, X. Wang, Asymptotic enumeration of sparse 0-1 matrices with irregular row and column sums, J. Combin. Theory Ser. A, 113 (2006) 291-324]. We also generalise the enumeration to matrices over any subset of the nonnegative integers that includes 0 and 1.