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

Let s = (s1, . . . , sm) and t = (t1, . . . , tn) be vectors of nonnegative integer-valued functions of m,n with equal sum S = ∑m i=1 si = ∑n j=1 tj. Let M(s, t) be the number of m × n matrices with nonnegative integer entries such that the ith row has row sum si and the jth column has column sum tj 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 = maxi si and t = maxj tj. 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 (Canfield and McKay, 2007). In this paper we extend the sparse range to the case st = o(S2/3). The proof in part follows a previous asymptotic enumeration of 0-1 matrices under the same conditions (Greenhill, McKay and Wang, 2006). We also generalise the enumeration to matrices over any subset of the nonnegative integers that includes 0 and 1.