Classical patterns in Mallows permutations

2024/10/22 by Victor Dubach, Dubach, Victor
Mathematics · #05A05 #60C05 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2410.17228

openalex publication_date 2024/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study classical pattern counts in Mallows random permutations with parameters (n,q_n), as n→∞. We focus on three different regimes for the parameter q = q_n. When n^3/2(1-q)→0, 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→1 and n(1-q)→∞, we use results on the displacements of permutation points to find the order of magnitude of pattern counts. When q∈(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.

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