The probability of avoiding consecutive patterns in the Mallows distribution

TL;DR

This paper investigates the probability that permutations drawn from the Mallows distribution avoid certain consecutive patterns, establishing growth rates, extending combinatorial methods, and analyzing the distribution of pattern occurrences.

Contribution

It proves the existence of growth rates for pattern avoidance probabilities under the Mallows distribution and extends the cluster method to account for inversions, providing new analytical tools.

Findings

Growth rates exist for all patterns and positive q values.

Extended cluster method tracks inversions in pattern avoidance.

Pattern occurrence counts are approximately normal under certain conditions.

Abstract

We use various combinatorial and probabilistic techniques to study growth rates for the probability that a random permutation from the Mallows distribution avoids consecutive patterns. The Mallows distribution behaves like a $q$-analogue of the uniform distribution by weighting each permutation $\pi$ by $q^{inv(\pi)}$, where $inv(\pi)$ is the number of inversions in $\pi$ and $q$ is a positive, real-valued parameter. We prove that the growth rate exists for all patterns and all $q>0$, and we generalize Goulden and Jackson's cluster method to keep track of the number of inversions in permutations avoiding a given consecutive pattern. Using singularity analysis, we approximate the growth rates for length-3 patterns, monotone patterns, and non-overlapping patterns starting with 1, and we compare growth rates between different patterns. We also use Stein's method to show that, under certain assumptions on $q$, the length of $\sigma$, and $inv(\sigma)$, the number of occurrences of a given pattern $\sigma$ is well approximated by the normal distribution.