Enumerating regions of Shi arrangements per Weyl Cone

Given a Shi arrangement ${\mathcal{A}}_{\mathrm{\Phi }}$, it is well-known that the total number of regions is counted by the parking number of type $\mathrm{\Phi }$ and the total number of regions in the dominant cone is given by the Catalan number of type $\mathrm{\Phi }$. In the case of the latter, Shi gave a bijection between antichains in the root poset of $\mathrm{\Phi }$ 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 ${\mathcal{A}}_{\mathrm{\Phi }}$ 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.