Colored Eulerian Polynomials and the Colored Permutohedron

TL;DR

This paper introduces a colored generalization of Eulerian polynomials via a new colored permutohedron and a novel descent concept for colored permutations, providing combinatorial and geometric insights.

Contribution

It develops the $ ext{alpha}$-colored Eulerian polynomials and the $ ext{alpha}$-colored permutohedron, extending classical concepts to a colored setting.

Findings

Computed the $ ext{alpha}$-colored Eulerian polynomials using the $h$-vector of the colored permutohedron.

Defined a new notion of descent for colored permutations.

Established combinatorial and geometric interpretations of the colored Eulerian polynomials.

Abstract

This paper introduces a colored generalization of the Eulerian polynomials, denoted the $\alpha$-colored Eulerian polynomials. We first compute these polynomials by taking the $h$-vector of the $\alpha$-colored permutohedron, a colored analog of the permutohedron which we develop. We also arrive at the $\alpha$-colored Eulerian polynomials combinatorially by defining a new notion of descent for colored permutations.