Mixed volumes of hypersimplices
type: Item
instance of: scholarly article
title: Mixed volumes of hypersimplices
zbMATH Open document ID: 1344.05008
author: Gaku Liu
published in: The Electronic Journal of Combinatorics
publication date: 2016-09-13T00:00:00Z
full work available at URL: v23i3p19
review text: 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.
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11B68
zbMATH DE Number: 6626783
zbMATH Keywords: hypersimplex
zbMATH Keywords: mixed volume
zbMATH Keywords: Eulerian numbers
MaRDI profile type: Publication
cites work: The number of roots of a system of equations
cites work: Mixed volumes and slices of the cube
cites work: Permutohedra, Associahedra, and Beyond
cites work: Q4132537
cites work: Two combinatorial applications of the Aleksandrov-Fenchel inequalities
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description: scientific article; zbMATH DE number 6626783