Continuously Increasing Subsequences of Random Multiset Permutations

TL;DR

This paper estimates the expected length of the longest increasing subsequences in random permutations of multisets, confirming conjectures and revealing asymptotic behaviors as parameters grow large.

Contribution

It provides asymptotic estimates for the expected longest increasing subsequences in multiset permutations, confirming a conjecture and extending understanding of such structures.

Findings

Expected length of longest increasing subsequence $L()$ asymptotically equals $m$ for large $n$ and $m$.

Expected maximum length $L()$ asymptotically matches the inverse gamma function $(n)$ for large $n$ and $m$.

Confirmed a conjecture by Diaconis, Graham, He, and Spiro regarding multiset permutations.

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.