Whitney Numbers for Poset Cones

Abstract

Hyperplane arrangements dissect $\mathbb{R}^n$ 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 reflecting hyperplanes $x_i=x_j$ inside $\mathbb{R}^n$ for the symmetric group, thought of as the type $A_{n-1}$ 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 left-to-right 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.