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：序列$\pi_1,\pi_2,\dots$如果满足每个排列$\sigma$在$\pi_n$中的诱导密度当$n\to\infty$时收敛于$1/|\sigma|!$，则称其为“拟随机的”。我们证明$\pi_1,\pi_2,\dots$是拟随机的当且仅当每个排列$\sigma$在集合$$\{123,321,2143,3412,2413,3142\}$$中的密度收敛于$1/|\sigma|!$。此前，具有此性质的最小基数集合（称为“拟随机强制”集合）已知介于四到八之间。事实上，我们证明存在一个由这六个排列密度构成的线性表达式可以强制拟随机性，并证明了在线性表达式中正系数排列密度的意义上这是最优的。在理论统计学语言中，该表达式提供了一种与斯皮尔曼$\rho$相关的双变量连续分布非参数独立性检验新方法。