Descents on quasi-Stirling permutations

Abstract

Stirling permutations were introduced by Gessel and Stanley in 1978, who enumerated them by the number of descents to give a combinatorial interpretation of certain polynomials related to Stirling numbers. A natural extension of these permutations are quasi-Stirling permutations, which can be viewed as labeled noncrossing matchings. They were recently introduced by Archer et al., motivated by the fact that the Koganov–Janson correspondence between Stirling permutations and labeled increasing plane trees extends to a bijection between quasi-Stirling permutations and the same set of trees without the increasing restriction.In this paper we prove a conjecture of Archer et al. stating that there are (n+1)n−1 quasi-Stirling permutations of size n having n descents. More generally, we give the generating function for quasi-Stirling permutations by the number of descents, expressed as a compositional inverse of the generating function of Eulerian polynomials. We also find the analogue for quasi-Stirling permutations of the main result from Gessel and Stanley's paper. We prove that the distribution of descents on these permutations is asymptotically normal, and that the roots of the corresponding quasi-Stirling polynomials are all real, in analogy to Bóna's results for Stirling permutations.Finally, we generalize our results to a one-parameter family of permutations that extends k-Stirling permutations, and we refine them by also keeping track of the number of ascents and the number of plateaus.