Context-free Grammars and Multivariate Stable Polynomials over Stirling Permutations

Abstract: Recently, Haglund and Visontai established the stability of the multivariate Eulerian polynomials as the generating polynomials of the Stirling permutations, which serves as a unification of some results of B\'{o}na, Brenti, Janson, Kuba, and Panholzer concerning Stirling permutations. Let $B_n(x)$ be the generating polynomials of the descent statistic over Legendre-Stirling permutations, and let $T_n(x)=2^nC_n(x/2)$, where $C_n(x)$ are the second-order Eulerian polynomials. Haglund and Visontai proposed the problems of finding multivariate stable refinements of the polynomials $B_n(x)$ and $T_n(x)$. We obtain context-free grammars leading to multivariate stable refinements of the polynomials $B_n(x)$ and $T_n(x)$. Moreover, the grammars enable us to obtain combinatorial interpretations of the multivariate polynomials in terms of Legendre-Stirling permutations and marked Stirling permutations. Such stable multivariate polynomials provide solutions to two problems posed by Haglund and Visontai.