Faces of generalized permutohedra

Abstract

The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and γ- 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 γ-vectors involving descent statistics. This includes a combinatorial interpretation for γ-vectors of a large class of generalized permutohedra which are flag simple polytopes, and confirms for them Gal's conjecture on the nonnegativity of γ-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 New- comb's problem. We give (and conjecture) upper and lower bounds for f-, h-, and γ-vectors within several classes of generalized permutohedra. An appendix discusses the equivalence of various notions of deformations of simple polytopes.