Whitney Numbers for Poset Cones

  title={Whitney Numbers for Poset Cones},
  author={Galen Dorpalen-Barry and Jang Soo Kim and Victor Reiner},
Hyperplane arrangements dissect ℝ n $\mathbb {R}^{n}$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem generalizes to count chambers within any cone defined as the intersection of a collection of halfspaces from the arrangement, leading to a notion of Whitney numbers for each cone. This paper focuses on cones within the braid arrangement, consisting of the… 
The Varchenko-Gel'fand Ring of a Cone
For a hyperplane arrangement in a real vector space, the coefficients of its Poincaré polynomial have many interpretations. An interesting one is provided by the Varchenko-Gel’fand ring, which is the
Gorenstein braid cones and crepant resolutions
To any poset P , we associate a convex cone called a braid cone. We also associate a fan and study the toric varieties the cone and fan define. The fan always defines a smooth toric variety XP ,
Enumeration of Gelfand-Cetlin type reduced words
The combinatorics of reduced words and commutation classes plays an important role in geometric representation theory. A string polytope is a lattice polytope associated to each reduced word of the
Labeled sample compression schemes for complexes of oriented matroids
It is shown that the topes of a complex of oriented matroids (abbreviated COM) of VC-dimension d admit a proper labeled sample compression scheme of size d, which is a step towards the sample compression conjecture.
Valuations and the Hopf Monoid of Generalized Permutahedra
We prove that the Hopf structure on generalized permutahedra descends, modulo the inclusion-exclusion relations, to an indicator Hopf monoid of generalized permutahedra that is isomorphic to the Hopf


On the Charney-Davis and Neggers-Stanley conjectures
It is shown that the sphere is flag whenever the poset P has width at most 2, and it is speculated that the proper context in which to view both of these conjectures may be the theory of Koszul algebras, and some evidence is presented.
Quotients of coxeter complexes and p-partitions
Let $W$ be a finite reflection group acting on R$\sp n$. As $W$ preserves the unit sphere S$\sp{n-1}$, for any subgroup $G\ \subseteq\ \ W$, there is a quotient S$\sp{n-1} /G$ of this sphere under
COMs: Complexes of oriented matroids
This work describes a binary composition scheme by which every COM can successively be erected as a certain complex of oriented matroids, in essentially the same way as a lopsided set can be glued together from its maximal hypercube faces.
The Varchenko Matrix for Cones
Consider an arrangement of hyperplanes and assign to each hyperplane a weight. By using this weights Varchenko defines a bilinear form on the vector space freely generated by the regions of the
A combinatorial analysis of topological dissections
Abstract From a topological space remove certain subspaces (cuts), leaving connected components (regions). We develop an enumerative theory for the regions in terms of the cuts, with the aid of a
Principal Γ-cone for a tree
For any tree G, we introduce G-cones consisting of chambers and enumerate the number of chambers contained in two particular (called principal) G-cones. The problem is equivalent to the combinatorial
Faces of Generalized Permutohedra
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,
q-Hook length formulas for forests
Two q-analogues of a hook length formula of Knuth for the number of linear extensions of a partially ordered set whose Hasse diagram is a rooted forest give formulas for the inversion index and the major index generating functions over permutations which correspond to linear extensionsof a labeled forest.
Permutation statistics and linear extensions of posets
The results extend and unify results of MacMahon, Foata, and Schutzenberger, and the authors, and explore classes of permutations which are invariant under Foata's bijection.
Generalized quotients in Coxeter groups
For (W, S) a Coxeter group, we study sets of the form W/V = {w E W I l(wv) = 1(w) + I(v) for all v E V}, where V C W. Such sets W/V, here called generalized quotients, are shown to have much of the