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:$1,\\ pi_ 1,\ pi_ 2,\ dots美元的排列顺序据说是 $123, 321, 21443, 3412, 2413, 3142 $美元 的每套调整方式的密度要合为 $\ pi_n$\ gma_n$ 美元, 美元是 $_\ pi_ 1,\ pi_ 2,\ dots$ 美元 。 我们证明$1, 1,\ pi_ 2, 美元是 准兰度。 如果每套调整方式的密度为 123, 321, 21443, 3412, 2413, 3142 $ 美元 的密度要合于 $/ {sigma_ $! 美元 。 以前, 使用此属性的一组最小基点, 称为 “ quasirranom- fornation” 。 我们证明这组六种变式的密度有一个单一线性表示, 并显示这是最好的可能, 因为它没有更短的线性表示 $ perloinalational adationalationalation denationalation denal deal deal deal dealtistrislationalitalitalitalslational estalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalitalital</s>