Classical patterns in Mallows permutations

TL;DR

This paper investigates the distribution of classical pattern counts in Mallows permutations across different parameter regimes, establishing central limit theorems and asymptotic behaviors using coupling, regenerative properties, and $U$-statistics.

Contribution

It provides new asymptotic results and CLTs for pattern counts in Mallows permutations under various parameter regimes, extending understanding beyond uniform permutations.

Findings

Pattern counts follow a CLT similar to uniform permutations when $n^{3/2}(1-q)\to0$.

Order of magnitude of pattern counts determined when $q\to1$ and $n(1-q)\to\infty$.

Pattern counts relate to $U$-statistics and satisfy CLTs for fixed $q$, with a constructed Mallows process satisfying a functional CLT.

Abstract

We study classical pattern counts in Mallows random permutations with parameters $(n,q_n)$, as $n\to\infty$. We focus on three different regimes for the parameter $q = q_n$. When $n^{3/2}(1-q)\to0$, we use coupling techniques to prove that pattern counts in Mallows random permutations satisfy a central limit theorem with the same asymptotic mean and variance as in uniformly random permutations. When $q\to1$ and $n(1-q)\to\infty$, we use results on the displacements of permutation points to find the order of magnitude of pattern counts. When $q\in(0,1)$ is fixed, we use the regenerative property of the Mallows distribution to compare pattern counts with certain $U$-statistics, and establish central limit theorems. We also construct a specific Mallows process, that is a coupling of Mallows distributions with $q$ ranging from $0$ to $1$, for which the process of pattern counts satisfies a functional central limit theorem.