The Ascent-Plateau Statistics on Stirling Permutations

Yeong-Nan Yeh,Shi-Mei Ma,Jun Ma

doi:10.37236/8008

Abstract

A permutation $\sigma$ of the multiset $\{1,1,2,2,\ldots,n,n\}$ is called a Stirling permutation of order $n$ if $\sigma_s>\sigma_i$ as long as $\sigma_i=\sigma_j$ and $i<s <j$. In this paper, we present a unified refinement of the ascent polynomials and the ascent-plateau polynomials of Stirling permutations. In particular, by using Foata and Strehl's group action, we prove that the pairs of statistics (left ascent-plateau, ascent) and (left ascent-plateau, plateau) are equidistributed over Stirling permutations of given order, and we show the $\gamma$-positivity of the enumerative polynomial of left ascent-plateaus, double ascents and descent-plateaus. A connection between the $\gamma$-coefficients of this enumerative polynomial and Eulerian numbers is also established.

Highlights

A Stirling permutation of order n is a permutation of the multiset {1, 1, 2, 2, . . . , n, n} such that for each i, 1 i n, all entries between the two occurrences of i are larger than i.Denote by Qn the set of Stirling permutations of order n
We identify an n-Stirling permutation σ1σ2 · · · σ2n with the word σ0σ1σ2 · · · σ2n, where σ0 = 0
We introduce the Foata-Strehl action on Stirling permutations by φi(σ) =

Summary

Introduction

A Stirling permutation of order n is a permutation of the multiset {1, 1, 2, 2, . . . , n, n} such that for each i, 1 i n, all entries between the two occurrences of i are larger than i. Let des (σ), asc (σ) and plat (σ) denote the numbers of descents, ascents and plateaus of σ, respectively. A classical result of Bona [2] says that descents, ascents and plateaus are equidistributed, i.e., xdes σ =. This equidistributed result have been extensively studied by Janson, Kuba, Panholzer, Haglund, Visontai, Chen, Fu et al, see [5, 8, 9, 10] and references therein. An occurrence of a left ascent-plateau is an index i such that σi−1 < σi = σi+1, where i ∈ {1, 2, . Let dasc (σ) and dp (σ) denote the numbers of double ascents and descent-plateaus of σ, respectively. The main tools of the proofs are the grammatical technique and a variation of the Foata and Strehl’s group action

Main results

A grammatical labeling of Stirling permutations

Equidistributed statistics

Connection with Eulerian numbers

Findings

Concluding remarks