Mixed Eulerian numbers and beyond

TL;DR

This work derives explicit, non-recursive formulas for classical mixed Eulerian numbers $A(a_1,\dots,a_n)$ by interpreting them as coefficients of a volume polynomial $P$ in the permutohedral Chow ring, and extends the framework to matroidal mixed Eulerian numbers $A_M(a_1,\dots,a_n)$ via intersection theory with divisors $L_i$. It establishes that the invariant generated by matroidal mixed Eulerian numbers is equivalent to Derksen's $\mathcal{G}$-invariant, thereby expressing the Tutte polynomial in terms of these numbers and showing universality for valuative invariants. The paper develops both explicit and recursive formulas, including two recursive schemes for computing $A_M$, and situates these results within the geometry of the permutohedral variety $X_{\Pi_n}$ and its equivariant Chow ring, bridging combinatorics, matroid theory, and algebraic geometry. Overall, it provides a concrete, computable link between matroid invariants and classical polynomial invariants, with broad implications for understanding mixed volumes, Eulerian-type sequences, and their algebraic-geometric underpinnings.

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.

Theorems & Definitions (59)

• Proposition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Corollary 1.6
• Definition 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10