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

Abstract

Let s = ( s 1 , … , s m ) and t = ( t 1 , … , 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.