Between Shi and Ish |
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Authors: | Rui Duarte António Guedes de Oliveira |
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Affiliation: | 1. CIDMA and Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal;2. CMUP and Department of Mathematics, Faculty of Sciences, University of Porto, 4169-007 Porto, Portugal |
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Abstract: | We introduce a new family of hyperplane arrangements in dimension that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions – the connected components of the complement of the union of the hyperplanes – which can be bijectively labeled with the Pak–Stanley labeling. In addition, we show that, in the cases of the Shi and the Ish arrangements, the number of labels with reverse centers of a given length is equal, and conjecture that the same happens with all of the members of the family. |
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Keywords: | Hyperplane arrangement Shi arrangement Ish arrangement Pak-Stanley labeling Parking functions |
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