Faces of Generalized Permutohedra

TL;DR

This paper computes face numbers and vector properties of generalized permutohedra, including explicit formulas and combinatorial interpretations, confirming conjectures and exploring relations with classical combinatorial numbers.

Contribution

It provides explicit formulas, combinatorial interpretations, and bounds for face and vector counts of generalized permutohedra, extending understanding of their structure.

Findings

Confirmed Gal's conjecture for flag simple polytopes.

Derived explicit formulas for h- and gamma-vectors.

Connected face vectors with Narayana numbers and Simon Newcomb's problem.

Abstract

The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and gamma-vectors. These polytopes include permutohedra, associahedra, graph-associahedra, simple graphic zonotopes, nestohedra, and other interesting polytopes. We give several explicit formulas for h-vectors and gamma-vectors involving descent statistics. This includes a combinatorial interpretation for gamma-vectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal's conjecture on nonnegativity of gamma-vectors. We calculate explicit generating functions and formulae for h-polynomials of various families of graph-associahedra, including those corresponding to all Dynkin diagrams of finite and affine types. We also discuss relations with Narayana numbers and with Simon Newcomb's problem. We give (and conjecture) upper and lower bounds for f-, h-, and gamma-vectors within several classes of generalized permutohedra. An appendix discusses the equivalence of various notions of deformations of simple polytopes.