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The $ \mathbf{s}$-Eulerian polynomials have only real roots


Authors: Carla D. Savage and Mirkó Visontai
Journal: Trans. Amer. Math. Soc. 367 (2015), 1441-1466
MSC (2010): Primary 05A05, 26C10; Secondary 05A19, 05A30
DOI: https://doi.org/10.1090/S0002-9947-2014-06256-9
Published electronically: October 10, 2014
MathSciNet review: 3280050
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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 $ \mathbf {s}$ of positive integers, which they called $ \mathbf {s}$-inversion sequences.

Our object of study is the generating polynomial of the ascent statistic over the set of $ \mathbf {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 $ \mathbf {s}$-Eulerian polynomial. The main result of this paper is that, for any sequence $ \mathbf {s}$ of positive integers, the $ \mathbf {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.


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Additional Information

Carla D. Savage
Affiliation: Department of Computer Science, North Carolina State University, Raleigh, North Carolina 27695-8206
Email: savage@ncsu.edu

Mirkó Visontai
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Address at time of publication: Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Email: visontai@kth.se

DOI: https://doi.org/10.1090/S0002-9947-2014-06256-9
Received by editor(s): March 2, 2013
Received by editor(s) in revised form: July 19, 2013
Published electronically: October 10, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.