Permutohedra, associahedra, and beyond

TL;DR

This paper explores the combinatorial and geometric properties of permutohedra and related polytopes, providing formulas for their volumes, lattice points, and introducing mixed Eulerian numbers with connections to various combinatorial sequences.

Contribution

It introduces new formulas for volume and lattice point polynomials of these polytopes and defines mixed Eulerian numbers linking them to classical combinatorial numbers.

Findings

Derived multiple formulas for volume and lattice point polynomials.

Defined mixed Eulerian numbers and related them to known combinatorial quantities.

Extended results to arbitrary Weyl groups.

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.