# Gamma-positivity of variations of Eulerian polynomials

## TL;DR

This paper explores binomial-Eulerian polynomials, revealing their gamma-positivity, geometric interpretations, and q-analogs, thus extending classical properties of Eulerian polynomials through algebraic and combinatorial frameworks.

## Contribution

It introduces binomial-Eulerian polynomials, establishes their gamma-positivity, and develops q-analogs and symmetric function identities with geometric interpretations.

## Key findings

- Binomial-Eulerian polynomials are palindromic and unimodal.
- They possess gamma-positivity similar to Eulerian polynomials.
- The paper provides q-analogs and algebraic interpretations of these 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 Sch\"utzenberger 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-Sch\"utzenberger 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.