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The active bijection between regions and simplices in supersolvable arrangements of hyperplanes. (English) Zbl 1188.52025

The paper is written in terms of real arrangements of hyperplanes. The results of this paper generalize to oriented matroids. The main features of the active reorientation-to-basis for general oriented matroids is recalled from E. Gioan and M. Las Vergnas [Eur. J. Comb. 30, No. 8, 1868–1886 (2009; Zbl 1193.05053)]. The authors provide many examples.
Authors’ summary: “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 [R. Stanley, Möbius Algebras, Conf. Proc. Waterloo 1971, 77–79, 81–42 (1975; Zbl 0377.05013) and Algebra Univers. 2, 197–217 (1972; Zbl 0256.06002)]. 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.”

MSC:

52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
52C40 Oriented matroids in discrete geometry
05B35 Combinatorial aspects of matroids and geometric lattices
05A05 Permutations, words, matrices
06B20 Varieties of lattices
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