Counting Consecutive Pattern Matches in $\mathcal{S}_n(132)$ and $\mathcal{S}_n(123)$

TL;DR

This paper analyzes the distribution of consecutive patterns in 123-avoiding and 132-avoiding permutations, providing insights into their joint distributions and extending methods to more general cases.

Contribution

It introduces new methods for studying the distribution of consecutive patterns in pattern-avoiding permutations, including joint distributions and broader cases.

Findings

Distribution of 3-length consecutive pattern matches characterized.

Joint distributions of multiple patterns analyzed.

Extensions to more general pattern-avoiding cases discussed.

Abstract

In this paper, we study the distribution of consecutive patterns in the set of 123-avoiding permutations and the set of 132-avoiding permutations, that is, in $\mathcal{S}_n(123)$ and $\mathcal{S}_n(132)$. We first study the distribution of consecutive pattern $\gamma$-matches in $\mathcal{S}_n(123)$ and $\mathcal{S}_n(132)$ for each length 3 consecutive pattern $\gamma$. Then we extend our methods to study the joint distributions of multiple consecutive patterns. Some more general cases are discussed in this paper as well.