Enumerating regions of Shi arrangements per Weyl Cone
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912715622055936 |
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| author | Dermenjian, Aram Tzanaki, Eleni |
| author_facet | Dermenjian, Aram Tzanaki, Eleni |
| contents | Given a Shi arrangement $\mathcal{A}_Φ$, it is well-known that the total number of regions is counted by the parking number of type $Φ$ and the total number of regions in the dominant cone is given by the Catalan number of type $Φ$. In the case of the latter, Shi gave a bijection between antichains in the root poset of $Φ$ and the regions in the dominant cone. This result was later extended by Armstrong, Reiner and Rhoades where they gave a bijection between the number of regions contained in an arbitrary Weyl cone $C_w$ in $\mathcal{A}_Φ$ and certain subposets of the root poset. In this article we expand on these results by giving a determinental formula for the precise number of regions in $C_w$ using paths in certain digraphs related to Shi diagrams. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_01725 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Enumerating regions of Shi arrangements per Weyl Cone Dermenjian, Aram Tzanaki, Eleni Combinatorics 20F55, 52C35, 05C20, 05C38, 05C30, 05C22, 14N10 Given a Shi arrangement $\mathcal{A}_Φ$, it is well-known that the total number of regions is counted by the parking number of type $Φ$ and the total number of regions in the dominant cone is given by the Catalan number of type $Φ$. In the case of the latter, Shi gave a bijection between antichains in the root poset of $Φ$ and the regions in the dominant cone. This result was later extended by Armstrong, Reiner and Rhoades where they gave a bijection between the number of regions contained in an arbitrary Weyl cone $C_w$ in $\mathcal{A}_Φ$ and certain subposets of the root poset. In this article we expand on these results by giving a determinental formula for the precise number of regions in $C_w$ using paths in certain digraphs related to Shi diagrams. |
| title | Enumerating regions of Shi arrangements per Weyl Cone |
| topic | Combinatorics 20F55, 52C35, 05C20, 05C38, 05C30, 05C22, 14N10 |
| url | https://arxiv.org/abs/2309.01725 |