[Paper Review] Context-free Grammars and Multivariate Stable Polynomials over Stirling Permutations

William Y. C. Chen, Robert X. J. Hao+1 more|arXiv (Cornell University)|Aug 7, 2012

Advanced Combinatorial Mathematics17 references19 citations

TL;DR

This paper introduces context-free grammars that generate multivariate stable polynomials over Legendre-Stirling and marked Stirling permutations, providing combinatorial interpretations and solving two open problems posed by Haglund and Visontai regarding stable refinements of the second-order Eulerian polynomials $B_n(x)$ and $T_n(x)$. The key result is the proof of multivariate stability for these polynomials, which implies real-rootedness under diagonalization.

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 $B_n(x)$ 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.

Motivation & Objective

• To resolve two open problems posed by Haglund and Visontai concerning multivariate stable refinements of the second-order Eulerian polynomials $B_n(x)$ and $T_n(x)$.
• To develop context-free grammars that generate multivariate stable polynomials over Legendre-Stirling and marked Stirling permutations.
• To provide combinatorial interpretations of these multivariate polynomials via grammatical labeling.
• To establish the stability of the generating polynomials using differential operators and multiaffine polynomial techniques.

Proposed method

• The authors define a context-free grammar $G_n$ with labeled variables $x_i, y_i, z_i, u_i, v_i$ for $0 \leq i < n$, and $v_n$, with specific production rules that encode combinatorial structures of marked Stirling permutations.
• They use differential operators $D_k$ to iteratively apply the grammar rules, generating the multivariate polynomials $f_n = D_n D_{n-1} \cdots D_1(x_0)$.
• Stability is proven inductively by showing that each differential operator $D_k$ preserves stability of multiaffine polynomials over complex variables with positive imaginary parts.
• The proof relies on Lemma 5.1, which reduces stability verification to checking non-vanishing of a transformed polynomial $T(F)$ under complex variable substitutions.
• The multivariate polynomial $T_n(\mathbf{x},\mathbf{y},\mathbf{z})$ is shown to be stable by applying the same operator framework after appropriate variable specialization.
• Diagonalization and specialization techniques are applied to derive univariate real-rootedness results from multivariate stability.

Experimental results

Research questions

• RQ1 Can multivariate stable refinements of the generating polynomials $B_n(x)$ and $T_n(x)$ over Stirling permutations be constructed?
• RQ2 Do context-free grammars exist that generate such stable polynomials with combinatorial interpretations?
• RQ3 Is the multivariate polynomial $B_n(\mathbf{x},\mathbf{y},\mathbf{z},\mathbf{u},\mathbf{v})$ stable under complex variable conditions?
• RQ4 Can the stability of $T_n(\mathbf{x},\mathbf{y},\mathbf{z})$ be established using the same grammatical framework?
• RQ5 Does the multivariate stability imply real-rootedness of the univariate generating polynomials under specialization?

Key findings

• The multivariate polynomial $B_n(\mathbf{x},\mathbf{y},\mathbf{z},\mathbf{u},\mathbf{v})$ is proven to be stable, meaning it does not vanish when all variables have positive imaginary parts.
• The multivariate polynomial $T_n(\mathbf{x},\mathbf{y},\mathbf{z})$ is also shown to be stable, extending the stability result to the second-order Eulerian polynomial refinement.
• The context-free grammar framework successfully generates combinatorial interpretations of the polynomials in terms of Legendre-Stirling and marked Stirling permutations.
• Specialization of variables leads to the real-rootedness of the univariate generating polynomial $M_n(x)$, which counts barred descents in Legendre-Stirling permutations.
• The diagonalization $x_i = u_i = y_i = z_i = 1$, $v_i = x$ yields a real-rooted polynomial $M_n(x)$, confirming the real-rootedness of the descent-generating function over Legendre-Stirling permutations.
• The stability of $T_n(\mathbf{x},\mathbf{y},\mathbf{z})$ implies the real-rootedness of $C_n(x)$ under the specialization $x_i = z_i = 1$, $y_i = y$, recovering Bóna’s result.