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Given a Shi arrangement $\mathcal{A}_\Phi$, it is well-known that the totalnumber of regions is counted by the parking number of type $\Phi$ and the totalnumber 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 inthe root poset of $\Phi$ and the regions in the dominant cone. This result waslater extended by Armstrong, Reiner and Rhoades where they gave a bijectionbetween 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 weexpand on these results by giving a determinental formula for the precisenumber of regions in $C_w$ using paths in certain digraphs related to Shidiagrams.

Enumerating regions of Shi arrangements per Weyl Cone