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… 

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.

The Varchenko-Gel'fand ring of a cone

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 their commutation classes plays an important role in geometric representation theory. For a semisimple complex Lie group $G$, a string polytope is a convex

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



COMs: Complexes of oriented matroids

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

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,

Combinatorial Reciprocity Theorems

  • M. Beck
  • Mathematics
    Graduate Studies in Mathematics
  • 2018
A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting

q-Hook length formulas for forests

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