1234-avoiding permutations and Dyck paths

TL;DR

This paper introduces an injective map between 1234-avoiding permutations and pairs of Dyck paths, providing a new combinatorial characterization of these permutations and their structure.

Contribution

It defines a novel map $ u$ linking 1234-avoiding permutations to Dyck paths and characterizes its image, advancing combinatorial understanding of pattern-avoiding permutations.

Findings

The map $ u$ is injective on 1234-avoiding permutations.

Characterization of the image of $ u$ within pairs of Dyck paths.

New combinatorial insights into 1234-avoiding permutations.

Abstract

We define a map $\nu$ between the symmetric group $S_n$ and the set of pairs of Dyck paths of semilength $n$. We show that the map $\nu$ is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map.