The descent statistic over 123-avoiding permutations

TL;DR

This paper uses a bijection between 123-avoiding permutations and Dyck paths to analyze the descent distribution, revealing new combinatorial insights and exact Eulerian numbers for these permutations.

Contribution

It introduces a novel approach linking permutation descents to Dyck path features, providing explicit Eulerian number calculations for 123-avoiding permutations.

Findings

Descent statistics correspond to valleys and triple falls in Dyck paths.

Derived explicit Eulerian numbers for 123-avoiding permutations.

Established a combinatorial link between permutation descents and Dyck path features.

Abstract

We exploit Krattenthaler's bijection between 123-avoiding permutations and Dyck paths to determine the Eulerian distribution over the set $S_n(123)$ of 123-avoiding permutations in $S_n$. In particular, we show that the descents of a permutation correspond to valleys and triple falls of the associated Dyck path. We get the Eulerian numbers of $S_n(123)$ by studying the joint distribution of these two statistics on Dyck paths.