Counting Shi regions with a fixed separating wall

TL;DR

This paper calculates the number of dominant regions in the extended Shi arrangement of type A that have a specific hyperplane as a separating wall, extending the combinatorial understanding of Shi regions.

Contribution

It provides a formula for counting regions with a fixed separating wall in the extended Shi arrangement of type A, generalizing previous combinatorial results.

Findings

Derived an explicit count for regions with a fixed separating hyperplane.

Extended the combinatorial framework for Shi arrangements.

Connected the counting problem to existing combinatorial structures.

Abstract

Athanasiadis introduced separating walls for a region in the extended Shi arrangement and used them to generalize the Narayana numbers. In this paper, we fix a hyperplane in the extended Shi arrangement for type A and calculate the number of dominant regions which have the fixed hyperplane as a separating wall; that is, regions where the hyperplane supports a facet of the region and separates the region from the origin.