Classical patterns in Mallows permutations

We study classical pattern counts in Mallows random permutations with parameters (n,qn), as n→∞. We focus on three different regimes for the parameter q=qn. When n3∕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.