Mixed volumes of hypersimplices (Q311515)

Summary: In this paper we consider mixed volumes of combinations of hypersimplices. These numbers, called ``mixed Eulerian numbers'', were first considered by \textit{A. Postnikov} [Int. Math. Res. Not. 2009, No. 6, 1026--1106 (2009; Zbl 1162.52007)] 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.