## Abstract

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].

Original language | English |
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Pages (from-to) | 291-324 |

Number of pages | 34 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 113 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2006 |