Permutohedra, Associahedra, and Beyond

The volume and the number of lattice points of the permutohedron Pn are given by certain multivariate polynomials that have remarkable combinatorial properties. We give 3 different formulas for these polynomials. We also study 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 ConciniProcesi. These polytopes are constructed as Minkowsky sums of simplices. We calculate their volumes and describe their combinatorial structure. The coefficients of monomials in Vol Pn are certain positive integer numbers, which we call 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 (n+1) of trees (or parking functions), the binomial coefficients, etc. We calculate the mixed Eulerian numbers using certain binary trees. Many results are extended to an arbitrary Weyl group. 1. Permutohedron Let x1, . . . , xn+1 be real numbers. The permutohedron Pn(x1, . . . , xn+1) is the convex polytope in R defined as the convex hull of all permutations of the vector (x1, . . . , xn+1): Pn(x1, . . . , xn+1) := ConvexHull((xw(1), . . . , xw(n+1)) | w ∈ Sn+1), where Sn+1 is the symmetric group. Actually, this is an n-dimensional polytope that belongs to some hyperplane H ⊂ R. More generally, for a Weyl group W , we can define the weight polytope as a convex hull of a Weyl group orbit: PW (x) := ConvexHull(w(x) | w ∈ W ), where x is a point in the weight space on which the Weyl group acts. For example, for n = 2 and distinct x1, x2, x3, the permutohedron P2(x1, x2, x3) (type A2 weight polytope) is the hexagon shown below. If some of the numbers x1, x2, x3 are equal to each other then the permutohedron degenerates into a triangle, or even a single point. Date: November 21, 2004; based on transparencies dated July 26, 2004.