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Recently, Haglund and Visontai established the stability of the multivariateEulerian polynomials as the generating polynomials of the Stirlingpermutations, 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 overLegendre-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 theproblems of finding multivariate stable refinements of the polynomials $B_n(x)$and $T_n(x)$. We obtain context-free grammars leading to multivariate stablerefinements of the polynomials $B_n(x)$ and $T_n(x)$. Moreover, the grammarsenable us to obtain combinatorial interpretations of the multivariatepolynomials in terms of Legendre-Stirling permutations and marked Stirlingpermutations. Such stable multivariate polynomials provide solutions to twoproblems posed by Haglund and Visontai.

Context-free Grammars and Multivariate Stable Polynomials over Stirling Permutations