Decomposing Random Permutations into Order-Isomorphic Subpermutations

C. Groenland, Tom Johnston, D'aniel Kor'andi, Alexander Roberts, A. Scott, Jane Tan
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Abstract

Two permutations $s$ and $t$ are $k$-similar if they can be decomposed into subpermutations $s^1, \ldots, s^k$ and $t^1, \ldots, t^k$ such that $s^i$ is order-isomorphic to $t^i$ for all $i$. Recently, Dudek, Grytczuk and Ruci\'nski posed the problem of determining the minimum $k$ for which two permutations chosen independently and uniformly at random are $k$-similar. We show that two such permutations are $O(n^{1/3}\log^{11/6}(n))$-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalises to simultaneous decompositions of multiple permutations.
随机置换分解为序同构子置换
两个排列$s$和$t$是$k$-如果它们可以分解成子排列$s^1, \ldots, s^k$和$t^1, \ldots, t^k$,使得$s^i$对于所有$i$都是序同构于$t^i$。最近,Dudek, Grytczuk和Ruci 'nski提出了一个问题,即确定两个独立且均匀随机选择的排列的最小k$相似。我们证明了两个这样的排列是$O(n^{1/3}\log^{11/6}(n))$-与高概率相似,接近于多对数因子。我们的结果也推广到多重排列的同时分解。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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