zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Faces of generalized permutohedra. (English) Zbl 1167.05005

Authors’ 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 S. R. Gal’s conjecture [Discrete Comput. Geom. 34, No. 2, 269-284 (2005; Zbl 1085.52005)] 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 Newcomb’s problem. (Newcomb’s problem is the problem of counting permutations of a multiset with a given number of descents.)

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.

05A15Exact enumeration problems, generating functions
52B05Combinatorial properties of convex sets