Enumerating Descents on Quasi-Stirling Permutations and Plane Trees

Abstract

Gessel and Stanley introduced Stirling permutations to give a combinatorial interpretation of certain polynomials related to Stirling numbers. A natural extension of these permutations are quasi-Stirling permutations, which are in bijection with labeled rooted plane trees, and can be viewed as labeled noncrossing matchings. They were recently introduced by Archer et al., who conjectured that there are \((n+1)^{n-1}\) quasi-Stirling permutations of size n having n descents. Here we prove this conjecture. More generally, we enumerate quasi-Stirling permutations, as well as a one-parameter family that generalizes them, by the number of descents, giving an implicit equation for their generating function in terms of that of Eulerian polynomials. We also show that many of the properties of descents on usual permutations and on Stirling permutations have an analogue for quasi-Stirling permutations.