Mixed Eulerian Numbers and Peterson Schubert Calculus

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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 that 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.