Enumerating regions of Shi arrangements per Weyl cone

Given a Shi arrangement ShiΦ, 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, in Shi (1997), 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 in Armstrong et al. (2015) where they gave a bijection between the number of regions contained in an arbitrary Weyl cone Cw in ShiΦ 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 Cw using paths in certain digraphs related to Shi diagrams.