Gamma-positivity of variations of Eulerian polynomials
Abstract
An identity of Chung, Graham and Knuth involving binomial coefficients and Eulerian numbers motivates our study of a class of polynomials that we call binomial-Eulerian polynomials. These polynomials share several properties with the Eulerian polynomials. For one thing, they are $h$-polynomials of simplicial polytopes, which gives a geometric interpretation of the fact that they are palindromic and unimodal. A formula of Foata and Schützenberger shows that the Eulerian polynomials have a stronger property, namely $\gamma$-positivity, and a formula of Postnikov, Reiner and Williams does the same for the binomial-Eulerian polynomials. We obtain $q$-analogs of both the Foata-Schützenberger formula and an alternative to the Postnikov-Reiner-Williams formula, and we show that these $q$-analogs are specializations of analogous symmetric function identities. Algebro-geometric interpretations of these symmetric function analogs are presented.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2017
- DOI:
- 10.48550/arXiv.1702.06666
- arXiv:
- arXiv:1702.06666
- Bibcode:
- 2017arXiv170206666S
- Keywords:
-
- Mathematics - Combinatorics;
- 05A05;
- 05E05;
- 05E10;
- 05E45;
- 52B05
- E-Print:
- 30 pages