A sequence $\pi_1,\pi_2,\dots$ of permutations is said to be \emph{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$\pi_1,\pi_2,\dots$称为拟随机的，如果当$n\to\infty$时，每个置换$\sigma$在$\pi_n$中的诱导密度收敛于$1/|\sigma|!$。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，具备此性质的最小集合（称为“拟随机强制集”）的基数范围已知为四到八。事实上，我们表明存在一个由该集合中六种置换密度构成的线性表达式可强制拟随机性，并证明在具有正系数的置换密度线性表达式中，该表达式是最优的——即不存在更短的此类表达式具备该性质。在理论统计学语言中，该表达式为与斯皮尔曼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方法。