Mixed volumes of hypersimplices

TL;DR

This paper explores the combinatorial properties of mixed volumes of hypersimplices, called mixed Eulerian numbers, providing new interpretations and extending results to type B analogues.

Contribution

It offers a combinatorial interpretation of mixed Eulerian numbers and extends the theory to type B hypersimplices with new properties and interpretations.

Findings

Mixed Eulerian numbers enumerate certain permutations in S_n.

New combinatorial properties of mixed Eulerian numbers are established.

Type B analogues of mixed Eulerian numbers are introduced with interpretations.

Abstract

In this paper we consider mixed volumes of combinations of hypersimplices. These numbers, called "mixed Eulerian numbers", were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type $B$ analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers.