The Active Bijection between Regions and Simplices in Supersolvable Arrangements of Hyperplanes
Comparing two expressions of the Tutte polynomial of an ordered oriented matroid yields a remarkable numerical relation between the numbers of reorientations and bases with given activities. A natural activity preserving reorientation-to-basis mapping compatible with this relation is described in a series of papers by the present authors. This mapping, equivalent to a bijection between regions and no broken circuit subsets, provides a bijective version of several enumerative results due to Stanley, Winder, Zaslavsky, and Las Vergnas, expressing the number of acyclic orientations in graphs, or the number of regions in real arrangements of hyperplanes or pseudohyperplanes (i.e. oriented matroids), as evaluations of the Tutte polynomial. In the present paper, we consider in detail the supersolvable case – a notion introduced by Stanley – in the context of arrangements of hyperplanes. For linear orderings compatible with the supersolvable structure, special properties are available, yielding constructions significantly simpler than those in the general case. As an application, we completely carry out the computation of the active bijection for the Coxeter arrangements $A_n$ and $B_n$. It turns out that in both cases the active bijection is closely related to a classical bijection between permutations and increasing trees.