Enumerating regions of Shi arrangements per Weyl cone

Abstract

Given a Shi arrangement ${Shi}_{\Phi }$, it is well-known 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, in Shi (1997), 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 in Armstrong et al. (2015) where they gave a bijection between the number of regions contained in an arbitrary Weyl cone $C_w$ in ${Shi}_{\Phi }$ and certain subposets of the root poset. In this article we expand on these results by giving a determinantal formula for the precise number of regions in $C_w$ using paths in certain digraphs related to Shi diagrams.