We provide asymptotics for the poly-Bernoulli numbers, a combinatorial array that enumerates lonesum matrices. We obtain these (bivariate) asymptotics as an application of ACSV (Analytic Combinatorics in Several Variables). For the diagonal asymptotic (i.e., for the special case of square lonesum matrices) we present an alternative proof based on Parseval's identity. We also strengthen an existing result on asymptotic enumeration of permutations having a specified excedance set.

中文翻译：

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lonesum矩阵的渐近枚举

我们为多伯努利数提供渐近线，这是一个枚举孤和矩阵的组合数组。我们获得这些（双变量）渐近线作为 ACSV（几个变量中的分析组合）的应用。对于对角渐近（即平方孤和矩阵的特殊情况），我们提出了基于 Parseval 恒等式的替代证明。我们还加强了具有指定超出集的排列的渐近枚举的现有结果。