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 $\gamma$-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 the 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.