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arXiv · 2012.11451

On weak twins and up-and-down sub-permutations

Andrzej Dudek·Jarosław Grytczuk·Andrzej Ruciński
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Abstract

Two permutations $(x_1,\dots,x_w)$ and $(y_1,\dots,y_w)$ are weakly similar if $x_i \pi(i_2)<\pi(i_3)>...$ or $\pi(i_1)<\pi(i_2)>\pi(i_3)<...$. Let $\Pi_n$ be a random permutation selected uniformly from all $n!$ permutations of $[n]$. It is known that the length of a longest alternating permutation in $\Pi_n$ is asymptotically almost surely (a.a.s.) close to $2n/3$. We study the maximum length $\alpha(n)$ of a pair of disjoint alternating sub-permutations in $\Pi_n$ and show that there are two constants $1/3<c_1<c_2<1/2$ such that a.a.s. $c_1n\le \alpha(n)\le c_2n$. In addition, we show that the alternating shape is the most popular among all permutations of a given length.

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Andrzej Dudek, Jarosław Grytczuk, Andrzej Ruciński. 2020-12-21. On weak twins and up-and-down sub-permutations. https://arxiv.org/abs/2012.11451

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