We provide asymptotic formulae for the numbers of bipartite graphs with given degree sequence, and of loopless digraphs with given in- and out-degree sequences, for a wide range of parameters. Our results cover medium range densities and close the gaps between the results known for the sparse and dense ranges. In the case of bipartite graphs, these results were proved by Greenhill, McKay and Wang in 2006 and by Canfield, Greenhill and McKay in 2008, respectively. Our method also essentially covers the sparse range, for which much less was known in the case of loopless digraphs. For the range of densities which our results cover, they imply that the degree sequence of a random bipartite graph with m edges is accurately modelled by a sequence of independent binomial random variables, conditional upon the sum of variables in each part being equal to m. A similar model also holds for loopless digraphs.

中文翻译：

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按度数序列渐近枚举有向图和二部图

我们为具有给定度数序列的二部图和具有给定入度和出度序列的​​无环有向图的数量提供了渐近公式，适用于各种参数。我们的结果涵盖了中等范围的密度，并缩小了以稀疏和密集范围而闻名的结果之间的差距。在二部图的情况下，这些结果分别由 Greenhill、McKay 和 Wang 在 2006 年以及 Canfield、Greenhill 和 McKay 在 2008 年证明。我们的方法还基本上涵盖了稀疏范围，而在无环有向图的情况下，对此知之甚少。对于我们的结果涵盖的密度范围，它们意味着具有 m 条边的随机二分图的度数序列由一系列独立的二项式随机变量准确建模，条件是每个部分的变量总和等于 m。类似的模型也适用于无环有向图。