Every k entries in a permutation can have one of k! different relative orders, called patterns. How many times does each pattern occur in a large random permutation of size n? The distribution of this k!-dimensional vector of pattern densities was studied by Janson, Nakamura, and Zeilberger (2015). Their analysis showed that some component of this vector is asymptotically multinormal of order 1/sqrt(n), while the orthogonal component is smaller. Using representations of the symmetric group, and the theory of U-statistics, we refine the analysis of this distribution. We show that it decomposes into k asymptotically uncorrelated components of different orders in n, that correspond to representations of Sk. Some combinations of pattern densities that arise in this decomposition have interpretations as practical nonparametric statistical tests.

中文翻译：

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随机排列中的模式

排列中的每 k 个条目都可以有 k! 不同的相对顺序，称为模式。每个模式在大小为 n 的大随机排列中出现多少次？Janson、Nakamura 和 Zeilberger (2015) 研究了这种 k! 维模式密度向量的分布。他们的分析表明，该向量的某些分量是 1/sqrt(n) 阶的渐近多正态，而正交分量较小。使用对称群的表示和 U 统计理论，我们改进了对这种分布的分析。我们表明它分解为 n 中不同阶的 k 个渐近不相关的分量，对应于 Sk 的表示。在这种分解中出现的一些模式密度组合可以解释为实际的非参数统计检验。