Enumerating regions of Shi arrangements per Weyl Cone
Abstract
Given a Shi arrangement $\mathcal{A}_\Phi$, it is wellknown that the total number of regions is counted by the parking number of type $\Phi$ and the total number of regions in the dominant cone is given by the Catalan number of type $\Phi$. In the case of the latter, Shi gave a bijection between antichains in the root poset of $\Phi$ 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 $\mathcal{A}_\Phi$ 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2023
 DOI:
 10.48550/arXiv.2309.01725
 arXiv:
 arXiv:2309.01725
 Bibcode:
 2023arXiv230901725D
 Keywords:

 Mathematics  Combinatorics;
 20F55;
 52C35;
 05C20;
 05C38;
 05C30;
 05C22;
 14N10
 EPrint:
 30 pages, 9 figures