# Between Ish and Shi

## TL;DR

This paper introduces a new family of hyperplane arrangements encompassing Shi and Ish arrangements, demonstrating they share combinatorial properties like equal region counts and labeling characteristics, with conjectures extending these results.

## Contribution

It defines a new hyperplane arrangement family including Shi and Ish arrangements and proves they have identical region counts and labeling properties.

## Key findings

- All arrangements in the family have the same number of regions.
- The Pak-Stanley labeling bijectively labels the regions.
- Shi and Ish arrangements have equal counts of labels with reverse centers.

## Abstract

We introduce a new family of hyperplane arrangements in dimension $n\geq3$ that includes both the Shi arrangement and the Ish arrangement. We prove that all the members of a given subfamily have the same number of regions - the connected components of the complement of the union of the hyperplanes - which can be bijectively labeled with the Pak-Stanley labeling. In addition, we show that, in the cases of the Shi and the Ish arrangements, the number of labels with reverse centers of a given length is equal, and conjecture that the same happens with all of the members of the family.