The s-eulerian polynomials have only real roots

By C. Savage, Mirkó Visontai. 2012

109 citations recorded by the source corpus.

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 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, 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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/c04a9c5ed9ff0b8a8a525709599939b9294fef56

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Citations in the corpus

Unimodality problems in Ehrhart theory (2015, background): “The proof given by Savage and Visontai uses only the combinatorial interpretation of these polynomials in terms of ascent sequences andcompatible polynomialtechniques due to Chudnovsky and Seymour [25].”
ARITHMETIC ASPECTS OF SYMMETRIC EDGE POLYTOPES (2018, methodology): “In the light of Stanley’s Unimodality conjecture [33] of current great interest in Ehrhart theory are real-rooted h-polynomials and here recent success was made using interlacing polynomials techniques [2, 18, 30, 31].”
Gamma-positivity in combinatorics and geometry (2017, background): “Stembridge [128] (the proof that the W -Eulerian polynomials are real-rooted was completed more recently; see [101] and references therein).”