The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available.

中文翻译：

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改进了对给定行和列总和的非负整数矩阵数量的估计

具有给定行和列的非负整数矩阵的数量在数学和统计学的各种问题中具有特征，但没有已知的闭合形式表达式，因此我们依赖于近似值。在本文中，我们描述了一种新的近似，其动机是考虑具有非整数列数的矩阵的统计数据。该估计可以在矩阵大小的时间上进行线性评估，并返回在各种设置下与现有线性时间近似一样好或更好的精度结果。我们证明，该估计在稀疏表的情况下是渐近精确的，而根据经验，在密集表的情况下，该估计至少与其他线性时间估计一样好。我们还使用新的估计作为改进数值方法的起点，使用顺序重要性采样对给定边距的矩阵进行计数或采样。实现我们方法的代码可用。