Classical patterns in Mallows permutations
Victor Dubach

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 , as . We focus on three different regimes for the parameter . When , 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 and , we use results on the displacements of permutation points to find the order of magnitude of pattern counts. When is fixed, we use the regenerative property of the Mallows distribution to compare pattern counts with certain -statistics, and establish central limit theorems. We also construct a specific Mallows process, that is a coupling of Mallows distributions with ranging from to , for which the process of pattern counts satisfies a…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Mathematical Dynamics and Fractals · Advanced Mathematical Identities
