Whitney Numbers for Poset Cones

Hyperplane arrangements dissect $\mathbb {R}^{n}$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as sum nonnegative integers Whitney numbers the first kind. His generalizes to count within any cone defined intersection collection halfspaces from arrangement, leading notion for each cone. This paper focuses on cones braid consisting reflecting hyperplanes xi = xj inside symmetric group, thought type An− 1 reflection group. Here, We interpret this refinement all posets counting linear extensions according statistic that number left-to-right maxima permutation. When poset is disjoint union chains, we differently, using Foata’s theory cycle decomposition multiset permutations, simple generating function compiling these numbers.