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arXiv · 2110.10315

Continuously Increasing Subsequences of Random Multiset Permutations

Alexander Clifton·Bishal Deb·Yifeng Huang·Sam Spiro·Semin Yoo
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Abstract

For a word $\pi$ and integer $i$, we define $L^i(\pi)$ to be the length of the longest subsequence of the form $i(i+1)\cdots j$, and we let $L(\pi):=\max_i L^i(\pi)$. In this paper we estimate the expected values of $L^1(\pi)$ and $L(\pi)$ when $\pi$ is chosen uniformly at random from all words which use each of the first $n$ integers exactly $m$ times. We show that $\mathbb{E}[L^1(\pi)]\sim m$ if $n$ is sufficiently larger in terms of $m$ as $m$ tends towards infinity, confirming a conjecture of Diaconis, Graham, He, and Spiro. We also show that $\mathbb{E}[L(\pi)]$ is asymptotic to the inverse gamma function $\Gamma^{-1}(n)$ if $n$ is sufficiently large in terms of $m$ as $m$ tends towards infinity.

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Alexander Clifton, Bishal Deb, Yifeng Huang, Sam Spiro, Semin Yoo. 2021-10-20. Continuously Increasing Subsequences of Random Multiset Permutations. https://arxiv.org/abs/2110.10315

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