The Eulerian numbers on restricted centrosymmetric permutations

TL;DR

This paper investigates the distribution of descents in centrosymmetric permutations avoiding the pattern 123, using bijections with Dyck prefixes to analyze the case where the permutation length is even.

Contribution

It introduces a novel bijection between centrosymmetric pattern-avoiding permutations and Dyck prefixes, providing new insights into their descent distribution.

Findings

Established a bijection for even-length permutations avoiding 123

Analyzed descent distribution using Dyck prefix correspondence

Provided combinatorial enumeration results for these permutations

Abstract

We study the descent distribution over the set of centrosymmetric permutations that avoid the pattern of length 3. Our main tool in the most puzzling case, namely, $\tau=123$ and $n$ even, is a bijection that associates a Dyck prefix of length $2n$ to every centrosymmetric permutation in $S_{2n}$ that avoids 123.