Séminaire Lotharingien de Combinatoire, 89B.2 (2023), 12 pp.
Sergi Elizalde
The Distribution of Descents on Nonnesting Permutations
Abstract.
Motivated by recent results about descents on Stirling and quasi-Stirling permutations, we consider permutations of the multiset {1,1,2,2,...,n,n} that avoid the patterns 1221 and 2112. We call these nonnesting permutations, as they can be viewed as nonnesting matchings with labeled arcs.
We show that the polynomial describing the distribution of the number of descents 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.
Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.
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