2021/04/29 by Tatsuya Horiguchi, Horiguchi, Tatsuya · 1 citation
Mathematics · Chemistry · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Molecular spectroscopy and chirality
paper · pdf · doi:10.48550/arxiv.2104.14083
Let Φ be a root system. Postnikov introduced and studied the mixed Φ-Eulerian numbers. These numbers indicate the mixed volumes of Φ-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 Φ-Eulerian numbers when Φ is of type A. In this paper we connect a relation between mixed Φ-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 Φ-Eulerian numbers in arbitrary Lie types from Peterson Schubert calculus.