Continuously Increasing Subsequences of Random Multiset Permutations

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