Remixed Eulerian numbers

TL;DR

Remixed Eulerian numbers are a new polynomial q-deformation of mixed Eulerian numbers, exhibiting symmetry and unimodality, with combinatorial interpretations linked to weighted trees and permutahedral decompositions, expanding understanding of related polynomials.

Contribution

This paper introduces remixed Eulerian numbers as a novel q-deformation, providing new combinatorial interpretations and connecting them to existing polynomial families and geometric decompositions.

Findings

Proven symmetry and unimodality of the polynomials.

Established combinatorial interpretations via weighted trees and permutahedral decompositions.

Connected the new polynomials to known interpretations at q=1.

Abstract

Remixed Eulerian numbers are a polynomial $q$-deformation of Postnikov's mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as $q$-binomial coefficients and Garsia--Remmel's $q$-hit numbers. We study their combinatorics in more depth. As polynomials in $q$, they are shown to be symmetric and unimodal. By interpreting them as computing success probabilities in a simple probabilistic process we arrive at a combinatorial interpretation involving weighted trees. By decomposing the permutahedron into certain combinatorial cubes, we obtain a second combinatorial interpretation. At $q=1$, the former recovers Postnikov's interpretation whereas the latter recovers Liu's interpretation, both of which were obtained via methods different from ours.