The $\s$-Eulerian polynomials have only real roots
arXiv:1208.3831
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 . 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 -analogs. We also show that the MacMahon--Carlitz -Eulerian polynomial has only real roots whenever 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.
27 pages, revised version