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

Let s = ( s 1 , … , s m ) and t = ( t 1 , … , t n ) be vectors of non-negative integer-valued functions with equal sum S = ∑ i = 1 m s i = ∑ j = 1 n 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].