The Eulerian transformation

Abstract

Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation A : R [ t ] → R [ t ] \mathcal {A}: \mathbb {R}[t] \to \mathbb {R}[t] defined by A ( t n ) = A n ( t ) \mathcal {A}(t^n) = A_n(t) , where A n ( t ) A_n(t) denotes the n n -th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator A \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.