A Context-free Grammar for the $e$-Positivity of the Trivariate Second-order Eulerian Polynomials

TL;DR

This paper introduces a context-free grammar approach to analyze the $e$-positivity of trivariate second-order Eulerian polynomials, connecting combinatorial structures like trees and Stirling permutations.

Contribution

It develops a new grammatical framework that extends previous work on Eulerian polynomials to the trivariate case, providing tools for their $e$-positivity analysis.

Findings

Grammar-based computation of generating functions

Connection between Eulerian polynomials and combinatorial structures

Extension to trivariate second-order Eulerian polynomials

Abstract

Ma-Ma-Yeh made a beautiful observation that a transformation of the grammar of Dumont instantly leads to the $\gamma$-positivity of the Eulerian polynomials. We notice that the transformed grammar bears a striking resemblance to the grammar for 0-1-2 increasing trees also due to Dumont. The appearance of the factor of two fits perfectly in a grammatical labeling of 0-1-2 increasing plane trees. Furthermore, the grammatical calculus is instrumental to the computation of the generating functions. This approach can be adapted to study the $e$-positivity of the trivariate second-order Eulerian polynomials first introduced by Dumont in the contexts of ternary trees and Stirling permutations, and independently defined by Janson, in connection with the joint distribution of the numbers of ascents, descents and plateaux over Stirling permutations.