# Permutations avoiding a pattern of length three under Mallows distributions

## TL;DR

This paper analyzes the asymptotic probability that permutations avoiding certain length-three patterns occur under Mallows distributions, providing precise results for different parameter ranges and patterns.

## Contribution

It offers new asymptotic probability results for pattern avoidance in permutations under Mallows distributions, extending understanding across different parameter regimes.

## Key findings

- Precise asymptotic probabilities for pattern avoidance when q<1.
- Duality results for q>1 and patterns avoiding 123.
- Extension of pattern avoidance analysis to Mallows distributions.

## Abstract

We consider permutations avoiding a pattern of length three under the family of Mallows distributions. In particular, for any pattern $\tau\in S_3-\{321\}$, we obtain rather precise results on the asymptotic probability as $n\to\infty$ that a permutation $\sigma\in S_n$ under the Mallows distribution with parameter $q\in(0,1)$ avoids the pattern. By a duality between the parameters $q$ and $\frac1q$, we also obtain rather precise results on the above probability for $q>1$ and any pattern $\tau\in S_3-\{123\}$.