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Zbl 1167.05005
Postnikov, Alex; Reiner, Victor; Williams, Lauren
Faces of generalized permutohedra.
(English)
[J] Doc. Math., J. DMV 13, 207-273 (2008). ISSN 1431-0635; ISSN 1431-0643/e

Authors' 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. \par 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 {\it S. R. Gal}'s conjecture [Discrete Comput. Geom. 34, No. 2, 269-284 (2005; Zbl 1085.52005)] on the nonnegativity of $\gamma$-vectors. \par 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. (Newcomb's problem is the problem of counting permutations of a multiset with a given number of descents.) \par We give (and conjecture) upper and lower bounds for $f$-, $h$-, and $\gamma$-vectors within several classes of generalized permutohedra. \par An appendix discusses the equivalence of various notions of deformations of simple polytopes.
[Astrid Reifegerste (Magdeburg)]
MSC 2000:
*05A15 Combinatorial enumeration problems
52B05 Combinatorial properties of convex sets

Keywords: polytopes; face numbers; permutohedra; associahedra; graphic zonotopes; nestohedra; generating functions; Dynkin diagrams; Narayana numbers; Newcomb's problem

Citations: Zbl 1085.52005

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