Remixed Eulerian numbers: beyond the connected case

TL;DR

This paper introduces formulas for remixed Eulerian numbers, extending known results for special subfamilies and connecting them to various combinatorial sequences through a probabilistic approach.

Contribution

It provides new formulas for remixed Eulerian numbers for specific subfamilies, expanding the understanding of their combinatorial and algebraic properties.

Findings

Derived formulas for remixed Eulerian numbers in certain subfamilies

Extended known formulas for q-hit numbers by Garsia and Remmel

Connected remixed Eulerian numbers to classical combinatorial sequences

Abstract

In his study of generalised permutahedra, Postnikov considered the mixed volumes of hypersimplices, giving rise to the family of mixed Eulerian numbers. It comprises usual Eulerian numbers, binomial coefficients, Catalan numbers, and the large family of hit numbers. Nadeau and Tewari further gave a polynomial refinement of these, the remixed Eulerian numbers, which recover the natural $q$-analogs of these special families. Using a probabilistic model, we give several formulas for remixed Eulerian numbers for certain subfamilies, extending the known formulas for $q$-hit numbers due to Garsia and Remmel.