The Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials

TL;DR

This paper introduces the Dumont ansatz, a unified grammatical approach to study Eulerian, peak, and derivative polynomials, providing new combinatorial interpretations and identities.

Contribution

It develops a unified grammatical framework for various polynomial families, connecting them through transformations and offering new combinatorial insights.

Findings

Derived a convolution formula for peak polynomials leading to Gessel's formula.

Provided combinatorial interpretations for derivative polynomials.

Unified treatment of multiple polynomial families using the Dumont ansatz.

Abstract

We observe that three context-free grammars of Dumont can be brought to a common ground, via the idea of transformations of grammars, proposed by Ma-Ma-Yeh. Then we develop a unified perspective to investigate several combinatorial objects in connection with the bivariate Eulerian polynomials. We call this approach the Dumont ansatz. As applications, we provide grammatical treatments, in the spirit of the symbolic method, of relations on the Springer numbers, the Euler numbers, the three kinds of peak polynomials, an identity of Petersen, and the two kinds of derivative polynomials, introduced by Knuth-Buckholtz and Carlitz-Scoville, and later by Hoffman in a broader context. We obtain a convolution formula on the left peak polynomials, leading to the Gessel formula. In this framework, we are led to the combinatorial interpretations of the derivative polynomials due to Josuat-Verg\`es.