Mixed Eulerian numbers and Peterson Schubert calculus

TL;DR

This paper explores the connection between mixed Eulerian numbers associated with root systems and Peterson Schubert calculus, providing combinatorial models and simplified computations across various Lie types.

Contribution

It establishes a link between mixed $\

Findings

Provides a combinatorial model using left-right diagrams.

Derives simplified formulas for mixed Eulerian numbers in all Lie types.

Connects mixed Eulerian numbers with Peterson Schubert calculus.

Abstract

Let $\Phi$ be a root system. Postnikov introduced and studied the mixed $\Phi$-Eulerian numbers. These numbers indicate the mixed volumes of $\Phi$-hypersimplices. As specializations of these numbers, one can obtain the usual Eulerian numbers, the Catalan numbers, and the binomial coefficients. Recent work of Berget-Spink-Tseng gave a simple computation for the mixed $\Phi$-Eulerian numbers when $\Phi$ is of type $A$. In this paper we connect a relation between mixed $\Phi$-Eulerian numbers and Peterson Schubert calculus. By using the connection, we provide a combinatorial model for the computation of Berget-Spink-Tseng in terms of left-right diagrams which were introduced by Abe-Horiguchi-Kuwata-Zeng for the purpose of Peterson Schubert calculus. We also derive a simple computation for the mixed $\Phi$-Eulerian numbers in arbitrary Lie types from Peterson Schubert calculus.