Mixed Eulerian numbers and beyond

TL;DR

This paper derives explicit formulas for matroidal mixed Eulerian numbers, proving their equivalence to Derksen's -invariant and providing the first non-recursive formula for mixed Eulerian numbers using combinatorial methods.

Contribution

It introduces explicit formulas for matroidal mixed Eulerian numbers and establishes their equivalence to Derksen's -invariant, advancing the understanding of these combinatorial invariants.

Findings

Proved the equivalence of matroidal mixed Eulerian numbers and Derksen's -invariant.

Provided the first explicit, non-recursive formula for mixed Eulerian numbers.

Developed a combinatorial approach inspired by Schubert and Huh.

Abstract

We derive explicit formulas for the matroidal mixed Eulerian numbers. We resolve a question posed by Berget, Spink, and Tseng, demonstrating that the invariant defined by matroidal mixed Eulerian numbers is precisely equivalent to Derksen's $\mathcal{G}$-invariant. As an application, we provide the first explicit, non-recursive formula for mixed Eulerian numbers. Our combinatorial approach draws inspiration from the classical work of Schubert and incorporates the cutting-edge contributions of Huh.