  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.
