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Permutohedra, associahedra, and beyond. (English) Zbl 1162.52007

Consider a real $n$-dimensional vector space with a fixed basis. Let $x=\left({x}_{1},...,{x}_{n+1}\right)$ be an arbitrary point of this space. The permutohedron ${P}_{n}\left({x}_{1},...,{x}_{n+1}\right)$ is the convex polytope defined as the convex hull of the points whose coordinates are obtain from the coordinates of $x$ by permutations. Permutahedra appear in representation theory as weight polytopes of irreducible representations of $G{L}_{n}$ and in geometry as moment polytopes.

The volume and the number of lattice points of a permutohedron ${P}_{n}$ are given by certain multivariate polynomials that have remarkable combinatorial properties. The author introduces several different formulae for these polynomials. Additionally he studies a more general class of polytopes that includes the permutohedron, the associahedron, the cyclohedron, the Stanley-Pitman polytope, and various generalized associahedra related to wonderful compactifications of De Concini-Procesi. These polytopes are constructed as Minkowsky sums of simplices. The paper contains a calculation of their volumes and a description of their combinatorial structure. The coefficients of monomials in $Vol{P}_{n}$ are certain positive integers, which are called the mixed Eulerian numbers. These numbers are equal to the mixed volumes of hypersimplices. Various specializations of these numbers give the usual Eulerian numbers, the Catalan numbers, the numbers of trees (i.e. ${\left(n+1\right)}^{n-1}$), the binomial coefficients, etc. The author calculates the mixed Eulerian numbers using binary trees. Many results are extended to an arbitrary Weyl group.

##### MSC:
 52B20 Lattice polytopes (convex geometry) 52B11 $n$-dimensional polytopes