Context-free Grammars and Multivariate Stable Polynomials over Stirling Permutations

TL;DR

This paper develops context-free grammars to produce multivariate stable polynomials refining Stirling permutation generating functions, providing new combinatorial interpretations and solving open problems in the field.

Contribution

It introduces novel context-free grammars that generate multivariate stable polynomials related to Stirling permutations, advancing understanding of their combinatorial structure.

Findings

Established multivariate stability of refined Stirling permutation polynomials

Provided combinatorial interpretations via Legendre-Stirling and marked Stirling permutations

Solved open problems posed by Haglund and Visontai

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.