The Eulerian transformation

TL;DR

This paper explores the linear transformation defined by Eulerian polynomials, providing combinatorial, topological, and Ehrhart interpretations, and addressing properties like unimodality and real-rootedness, including disproving a longstanding conjecture.

Contribution

It introduces new interpretations of the Eulerian transformation and advances understanding of its algebraic and combinatorial properties, including disproving a conjecture and strengthening recent results.

Findings

Disproved Brenti's conjecture on real zeros preservation.

Provided new combinatorial and topological interpretations of the Eulerian transformation.

Extended recent results on binomial Eulerian polynomials.

Abstract

Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation $\mathcal{A} : \mathbb{R}[t] \to \mathbb{R}[t]$ defined by $\mathcal{A}(t^n) = A_n(t)$, where $A_n(t)$ denotes the $n$-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator $\mathcal{A}$, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials.