The active bijection between regions and simplices in supersolvable arrangements of hyperplanes
| Autor: | Emeric Gioan, Michel Las Vergnas |
|---|---|
| Přispěvatelé: | Algorithmes, Graphes et Combinatoire (ALGCO), Laboratoire d'Informatique de Robotique et de Microélectronique de Montpellier (LIRMM), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), Equipe combinatoire et optimisation (C&O), Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG), Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2006 |
| Předmět: |
Hyperplane arrangement
Context (language use) 0102 computer and information sciences [INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] 01 natural sciences Matroid permutation Theoretical Computer Science Combinatorics Permutation bijection basis [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] Discrete Mathematics and Combinatorics 0101 mathematics no broken circuit Mathematics hyperoctahedral arrangement Discrete mathematics Mathematics::Combinatorics region increasing tree Applied Mathematics activity 010102 general mathematics Coxeter group supersolvable braid arrangement oriented matroid Coxeter arrangement Oriented matroid Tutte polynomial Computational Theory and Mathematics Hyperplane 010201 computation theory & mathematics reorientation Bijection matroid Geometry and Topology |
| Zdroj: | The Electronic Journal of Combinatorics The Electronic Journal of Combinatorics, Open Journal Systems, 2006, 11 (2), pp.#R30 The Electronic Journal of Combinatorics, 2006, 11 (2), pp.#R30. ⟨10.37236/1887⟩ |
| ISSN: | 1077-8926 |
| DOI: | 10.37236/1887⟩ |
| Popis: | Dedicated to R. Stanley on the occasion of his 60th birthday (Stanley Festschrift); International audience; 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. |
| Databáze: | OpenAIRE |
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