Symmetric decompositions and Euler-Stirling statistics on Stirling permutations

Abstract

The Stirling permutations introduced by Gessel-Stanley have recently received considerable attention. Motivated by Ji's work on $(\alpha,\beta)$-Eulerian polynomials (Sci China Math., 2025) and Yan-Yang-Lin's work on $1/k$-Eulerian polynomials (J. Combin. Theory Ser. A, 2026), we present several symmetric decompositions of the enumerators related to Euler-Stirling statistics on Stirling permutations. Firstly, we provide a partial symmetric decomposition for the $1/k$-Eulerian polynomial. Secondly, we give several unexpected applications of the $(p,q)$-Eulerian polynomials, where $p$ marks the number of fixed points of permutations and $q$ marks that of cycles. From this paper, one can see that $(p,q)$-Eulerian polynomial contains a great deal of information about permutations and Stirling permutations. Using the change of grammars, we show that the $(\alpha,\beta)$-Eulerian polynomials introduced by Carlitz-Scoville can be deduced from the $(p,q)$-Eulerian polynomials by special parametrizations. We then introduce proper and improper ascent-plateau statistics on Stirling permutations. Moreover, we introduce proper ascent, improper ascent, proper descent and improper descent statistics on permutations. Furthermore, we consider the joint distributions of Euler-Stirling statistics on permutations, including the numbers of improper ascents, proper ascents, left-to-right minima and right-to-left minina. In the final part, we first give a symmetric decomposition of the joint distribution of the ascent-plateau and left ascent-plateau statistics, and then we show that the $q$-ascent-plateau polynomials are bi-$\gamma$-positive, where $q$ marks the number of left-to-right minima.