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 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 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 several existing results about the realrootedness 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, and partially settle a conjecture of Dilks, Petersen, Stembridge on type B affine Eulerian polynomials. It is then extended to several q-analogs. We 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 hyperoctahedral group and the wreath product groups, confirming further conjectures of Chow and Gessel, and Chow and Mansour, respectively.Our results have interesting geometric consequences as well.

Polyhedral geometry for lecture hall partitions

“…In particular, they show that h * (O(P, s); z) can be expressed as a sum of related polynomials which are an interlacing sequence. We should note that this is similar to the method used by Savage and Visontai [35] to prove Theorem 4.2. The reader should consult [7] for background and details on this method of proving real-rootedness.…”

“…The s-Eulerian polynomials also possess many of the same nice properties of Eulerian polynomials, namely unimodality and real-rootedness due to the following theorem of Savage and Visontai [35]. The proof of this theorem uses the idea of compatible polynomials and interlacing to prove real-rootedness.…”

On the descent polynomial of signed multipermutations

“…Savage and Visontai [14,Conjecture 3.25] further conjectured the following equidistribution, which was proved very recently (and independently) by Chen et al [4] using type B P -Partitions. t des(π) = 1 + 31t + 55t 2 + 9t 3 = e∈I (1,4,3,8) 4 t asc(e) .…”

Local h*-polynomials of some weighted projective spaces

“…They additionally showed that they contain the local h-polynomials for some well-studied subdivisions of simplices which are closely related to derangement polynomials. The h * -polynomials of the s-lecture hall simplices, called the s-Eulerian polynomials, were shown to be real-rooted and unimodal in [21]. The results of [16] then suggest that these nice properties of the h * -polynomial of a simplex can be inherited by its local h * -polynomial.…”

“…The simplices ∆ ! n and P s n are both reflexive (see [23] for the definition of reflexive) with h * -polynomial A n+1 (z), and the combinatorics of both are intimately tied to the combinatorics of inversion sequences [21,23]. Despite this, the distinction in their local h * -polynomials further highlights that they exhibit fundamentally different geometry and combinatorics.…”