The distribution of consecutive patterns of length 3 in $3\textrm{-}1\textrm{-}2$-avoiding permutations

TL;DR

This paper investigates the distribution of length-3 consecutive patterns in 3-1-2 pattern-avoiding permutations, revealing three equidistribution classes and establishing related statistical symmetries via Dyck path bijections.

Contribution

It introduces a novel analysis of consecutive pattern distributions in 3-1-2 avoiding permutations using Dyck path bijections and involutions, uncovering new equidistribution classes and statistical symmetries.

Findings

Consecutive patterns of length 3 split into 3 equidistribution classes.

An involution on Dyck paths explains the classes.

Equidistribution theorems for triplets of pattern-related statistics.

Abstract

We exploit Krattenthaler's bijection between the set $S_n(3\textrm{-}1\textrm{-}2)$ of permutations in $S_n$ avoiding the classical pattern $3\textrm{-}1\textrm{-}2$ and Dyck $n$-paths to study the distribution of every consecutive pattern of length 3 on the set $S_n(3\textrm{-}1\textrm{-}2)$. We show that these consecutive patterns split into 3 equidistribution classes, by means of an involution on Dyck paths due to E.Deutsch. In addition, we state equidistribution theorems concerning triplets of statistics relative to the occurrences of the consecutive patterns of length 3 in a permutation.