六个排列模式强制拟随机性
Six Permutation Patterns Force Quasirandomness
AI总结：
本文证明六个特定排列模式（123、321、2143、3412、2413、3142）的密度收敛即可强制排列序列的拟随机性，并给出最优的线性表达式，同时为二元连续分布提供新的非参数独立性检验。
摘要：
一个排列序列 $π_1,π_2,\dots$ 被称为“拟随机的”，如果每个排列 $σ$ 在 $π_n$ 中的诱导密度随着 $n\to\infty$ 收敛到 $1/|σ|!$。我们证明 $π_1,π_2,\dots$ 是拟随机的当且仅当集合 $$\{123,321,2143,3412,2413,3142\}$$ 中每个排列 $σ$ 的密度收敛到 $1/|σ|!$。此前，具有这一性质的集合的最小基数（称为“拟随机强制”集）已知在四到八之间。事实上，我们证明了该集合中六个排列的密度存在一个单一的线性表达式能够强制拟随机性，并证明这在以下意义下是最优的：不存在具有正系数的更短排列密度线性表达式具有这一性质。用理论统计学的语言来说，该表达式为与 Spearman 的 $ρ$ 相关的二元连续分布提供了一种新的非参数独立性检验。
英文摘要：
A sequence $π_1,π_2,\dots$ of permutations is said to be "quasirandom" if the induced density of every permutation $σ$ in $π_n$ converges to $1/|σ|!$ as $n\to\infty$. We prove that $π_1,π_2,\dots$ is quasirandom if and only if the density of each permutation $σ$ in the set $$\{123,321,2143,3412,2413,3142\}$$ converges to $1/|σ|!$. Previously, the smallest cardinality of a set with this property, called a "quasirandom-forcing" set, was known to be between four and eight. In fact, we show that there is a single linear expression of the densities of the six permutations in this set which forces quasirandomness and show that this is best possible in the sense that there is no shorter linear expression of permutation densities with positive coefficients with this property. In the language of theoretical statistics, this expression provides a new nonparametric independence test for bivariate continuous distributions related to Spearman's $ρ$.
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