# The Ascent-Plateau Statistics on Stirling Permutations

### 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.