Journal of Combinatorics

Volume 11 (2020)

Number 1

Gamma-positivity of variations of Eulerian polynomials

Pages: 1 – 33



John Shareshian (Department of Mathematics, Washington University, St. Louis, Missouri, U.S.A.)

Michelle L. Wachs (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)


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.


Eulerian polynomial, polytope, symmetric function

The first author was supported in part by NSF Grants DMS 1202337 and DMS 1518389.

The second author was supported in part by NSF Grants DMS 1202755 and DMS 1502606.

Received 7 February 2017

Published 27 September 2019