Whitney Numbers for Poset Cones
Abstract
Hyperplane arrangements dissect $\mathbb{R}^n$ into connected components called chambers, and a wellknown 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 reflecting hyperplanes $x_i=x_j$ inside $\mathbb{R}^n$ for the symmetric group, thought of as the type $A_{n1}$ reflection group. Here cones correspond to posets, chambers within the cone correspond to linear extensions of the poset, and the Whitney numbers of the cone interestingly refine the number of linear extensions of the poset. We interpret this refinement for all posets as counting linear extensions according to a statistic that generalizes the number of lefttoright maxima of a permutation. When the poset is a disjoint union of chains, we interpret this refinement differently, using Foata's theory of cycle decomposition for multiset permutations, leading to a simple generating function compiling these Whitney numbers.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1906.00036
 Bibcode:
 2019arXiv190600036D
 Keywords:

 Mathematics  Combinatorics;
 05Axx;
 06A07
 EPrint:
 Historical references added