The $\s$-Eulerian polynomials have only real roots

Abstract

We study the roots of generalized Eulerian polynomials via a novel approach. We interpret Eulerian polynomials as the generating polynomials of a statistic over inversion sequences. Inversion sequences (also known as Lehmer codes or subexcedant functions) were recently generalized by Savage and Schuster, to arbitrary sequences $\s$ of positive integers, which they called $\s$-inversion sequences. Our object of study is the generating polynomial of the {\em ascent} statistic over the set of $\s$-inversion sequences of length $n$. Since this ascent statistic over inversion sequences is equidistributed with the descent statistic over permutations we call this generalized polynomial the \emph{$\s$-Eulerian polynomial}. The main result of this paper is that, for any sequence $\s$ of positive integers, the $\s$-Eulerian polynomial has only real roots. This result is first shown to generalize many existing results about the real-rootedness of various Eulerian polynomials. We then show that it can be used to settle a conjecture of Brenti, that Eulerian polynomials for all finite Coxeter groups have only real roots. It is then extended to several $q$-analogs. We also show that the MacMahon--Carlitz $q$-Eulerian polynomial has only real roots whenever $q$ is a positive real number confirming a conjecture of Chow and Gessel. The same holds true for the $(\des,\finv)$-generating polynomials and also for the $(\des,\fmaj)$-generating polynomials for the hyperoctahedral group and the wreath product groups, confirming further conjectures of Chow and Gessel, and Chow and Mansour, respectively.