The geometry of a counting formula for deformations of the braid arrangement

TL;DR

This paper proves that for any deformation of the braid arrangement, each region contributes equally to the counting formula, providing a new geometric perspective and an alternative proof of the original enumeration.

Contribution

It removes the transitivity condition in Bernardi's formula, showing each region's contribution is always 1, and offers a new geometric interpretation of the counting formula.

Findings

Each region's contribution to the counting formula is 1.

Provides an alternative proof of Bernardi's counting formula.

Enhances understanding of the geometric structure of deformed braid arrangements.

Abstract

We consider real hyperplane arrangements whose hyperplanes are of the form $\{x_i - x_j = s\}$ for some integer $s$, which we call deformations of the braid arrangement. In 2018, Bernardi gave a counting formula for the number of regions of any deformation of the braid arrangement $\mathcal{A}$ as a signed sum over some decorated trees. He further showed that each of these decorated trees can be associated to a region $R$ of the arrangement $\mathcal{A}$, and hence we can consider the contribution of each region to the signed sum. Bernardi also implicitly showed that for transitive arrangements, the contribution of any region of the arrangement is $1$. We remove the transitivity condition, showing that for any deformation of the braid arrangement the contribution of a region to the signed sum is $1$. This provides an alternative proof of the original counting formula, and sheds light on the geometry underlying the formula. We further use this new geometric understanding to better understand the contribution of a tree.