# Consecutive patterns in restricted permutations and involutions

## TL;DR

This paper explores the combinatorial structures of restricted permutations and involutions, establishing bijections with Motzkin paths and analyzing pattern avoidance to derive distributions of permutation statistics.

## Contribution

It introduces a new bijection between restricted permutations and Motzkin paths, and uses it to analyze pattern avoidance and permutation statistics.

## Key findings

- Bijection between certain permutations and Motzkin paths.
- Distribution of des and inv statistics over specific pattern-avoiding sets.
- Enumeration of all length-three consecutive pattern distributions.

## Abstract

It is well-known that the set $\mathbf I_n$ of involutions of the symmetric group $\mathbf S_n$ corresponds bijectively - by the Foata map $F$ - to the set of $n$-permutations that avoid the two vincular patterns $\underline{123},$ $\underline{132}.$ We consider a bijection $\Gamma$ from the set $\mathbf S_n$ to the set of histoires de Laguerre, namely, bicolored Motzkin paths with labelled steps, and study its properties when restricted to $\mathbf S_n(1\underline{23},1\underline{32}).$ In particular, we show that the set $\mathbf S_n(\underline{123},{132})$ of permutations that avoids the consecutive pattern $\underline{123}$ and the classical pattern $132$ corresponds via $\Gamma$ to the set of Motzkin paths, while its image under $F$ is the set of restricted involutions $\mathbf I_n(3412).$ We exploit these results to determine the joint distribution of the statistics des and inv over $\mathbf S_n(\underline{123},{132})$ and over $\mathbf I_n(3412).$ Moreover, we determine the distribution in these two sets of every consecutive pattern of length three. To this aim, we use a modified version of the well-known Goulden-Jacson cluster method.