Descents on nonnesting multipermutations

TL;DR

This paper studies nonnesting permutations avoiding certain patterns, showing their descent distribution polynomial factors into well-known polynomials and is palindromic, with proofs via Dyck path transformations.

Contribution

It introduces nonnesting permutations, analyzes their descent distribution, and proves polynomial factorization and palindromicity using bijections and Dyck path involutions.

Findings

Descent distribution polynomial factors into Eulerian and Narayana polynomials.

The polynomial is palindromic, revealing symmetry.

Bijective proofs involve Dyck path transformations and Lalanne--Kreweras involution.

Abstract

Motivated by recent results on quasi-Stirling permutations, which are permutations of the multiset $\{1,1,2,2,\dots,n,n\}$ that avoid the "crossing" patterns 1212 and 2121, we consider nonnesting permutations, defined as those that avoid the patterns 1221 and 2112 instead. We show that the polynomial giving the distribution of the number of descents on nonnesting permutations is a product of an Eulerian polynomial and a Narayana polynomial. It follows that, rather unexpectedly, this polynomial is palindromic. We provide bijective proofs of these facts by composing various transformations on Dyck paths, including the Lalanne--Kreweras involution.