A sequence $\pi_1,\pi_2,\dots$ of permutations is said to be "quasirandom" if the induced density of every permutation $\sigma$ in $\pi_n$ converges to $1/|\sigma|!$ as $n\to\infty$. We prove that $\pi_1,\pi_2,\dots$ is quasirandom if and only if the density of each permutation $\sigma$ in the set $$\{123,321,2143,3412,2413,3142\}$$ converges to $1/|\sigma|!$. 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 $\rho$.

翻译：一个置换序列 $\pi_1,\pi_2,\dots$ 被称为“拟随机”的，如果对于每个置换 $\sigma$，其在 $\pi_n$ 中的诱导密度随着 $n\to\infty$ 收敛于 $1/|\sigma|!$。我们证明：$\pi_1,\pi_2,\dots$ 是拟随机的，当且仅当集合 $$\{123,321,2143,3412,2413,3142\}$$ 中每个置换 $\sigma$ 的密度收敛于 $1/|\sigma|!$。此前已知，具有该性质（称为“拟随机强制”性质）的集合的最小基数介于四到八之间。事实上，我们证明了存在一个关于该集合中六种置换密度的线性表达式，该表达式可强制拟随机性，并且这是最优的——即不存在具有正系数的更短置换密度线性表达式具有该性质。在理论统计学的术语中，该表达式为与斯皮尔曼 $\rho$ 相关的二元连续分布提供了一种新的非参数独立性检验。