Permutohedra, associahedra, and beyond
Alexander Postnikov
TL;DR
The paper develops a comprehensive framework to study volumes and lattice-point counts of the permutohedron and the broader class of generalized permutohedra, unifying three perspectives: Brion-type lattice-point formulas, Minkowski-sum/hypersimplex (mixed-volume) representations yielding mixed Eulerian numbers, and weight-polytope/Weyl-group descriptions. It introduces the dragon marriage condition and the deformation/cone approach to generalized permutohedra, links faces to a nested complex, and exposes numerous rich connections to Catalan-type counts, parking functions, and rooted trees. The work extends to root polytopes, Cayley-trick subdivisions, and a broad Ehrhart theory via generalized Todd operators, with concrete applications to shifted Young tableaux diagonals and multiple combinatorial structures such as graph associahedra and Pitman–Stanley polytopes. Collectively, the results provide versatile tools for computing volumes and lattice points across a wide spectrum of polytopes arising from combinatorics, geometry, and representation theory, and they reveal deep dualities and enumerative correspondences across Weyl groups and root systems.
Abstract
The volume and the number of lattice points of the permutohedron P_n are given by certain multivariate polynomials that have remarkable combinatorial properties. We give several different formulas for these polynomials. We also study a more general class of polytopes that includes the permutohedron, the associahedron, the cyclohedron, the Pitman-Stanley polytope, and various generalized associahedra related to wonderful compactifications of De Concini-Procesi. These polytopes are constructed as Minkowski sums of simplices. We calculate their volumes and describe their combinatorial structure. The coefficients of monomials in Vol P_n 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)^{n-1} of trees, the binomial coefficients, etc. We calculate the mixed Eulerian numbers using certain binary trees. Many results are extended to an arbitrary Weyl group.