Bijections between generalized Catalan families of types and |
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Authors: | Myrto Kallipoliti Eleni Tzanaki |
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Affiliation: | 1. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria;2. Department of Mathematics & Applied Mathematics, University of Crete, GR 70013, Heraklion, Greece |
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Abstract: | Motivated by the relation , holding for the -generalized Catalan numbers of type and , the connection between dominant regions of the -Shi arrangement of type and is investigated. More precisely, it is explicitly shown how copies of the set of dominant regions of the -Shi arrangement of type , biject onto the set of type such regions. This is achieved by exploiting two different viewpoints of the representative alcove of each region: the Shi tableau and the abacus diagram. In the same line of thought, a bijection between copies of the set of -Dyck paths of height and the set of lattice paths inside an rectangle is provided. |
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Keywords: | Shi hyperplane arrangements Abacus diagram Affine permutations Lattice paths Catalan numbers |
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