Context-free Grammars and Multivariate Stable Polynomials over Stirling Permutations
William Y.C. Chen, Robert X.J. Hao and Harold R.L. Yang
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 Bóna, Brenti, Janson, Kuba, and Panholzer concerning Stirling permutations. Let Bn(x) be the generating polynomials of the descent statistic over Legendre-Stirling permutations, and let Tn(x) = 2nCn(x/2), where Cn(x) are the second-order Eulerian polynomials. Haglund and Visontai proposed the problems of finding multivariate stable refinements of the polynomials Bn(x) and Tn(x). We obtain context-free grammars leading to multivariate stable refinements of the polynomials Bn(x) and Tn(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.
AMS Classification: 05A05, 05A15, 32A60, 68Q42
Keywords: context-free grammar, multivariate stable polynomial, stability preserving operator, Stirling permutation, Legendre-Stirling permutation