Enumerating regions of Shi arrangements per Weyl Cone

TL;DR

This paper provides a new determinantal formula to count the regions of Shi arrangements within specific Weyl cones, extending previous bijections and combinatorial results.

Contribution

It introduces a determinantal formula for counting regions in Weyl cones of Shi arrangements, expanding on prior bijections with root poset substructures.

Findings

Derived a determinantal formula for region counts in Weyl cones

Extended bijections between regions and subposets of root posets

Connected Shi diagram paths to region enumeration

Abstract

Given a Shi arrangement $\mathcal{A}_\Phi$, 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, 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.