The Bernardi formula for non-transitive deformations of the braid arrangement

TL;DR

This paper analyzes Bernardi's formula for counting regions in deformed braid arrangements, providing a combinatorial interpretation and an explicit enumeration for specific nested arrangements.

Contribution

It proves the sign contribution of boxed trees depends only on their underlying tree and introduces an algorithm for computing these contributions.

Findings

Each boxed tree set contributes 0, +1, or -1 to Bernardi's sum.

Constructs a sign-reversing involution for Ish-type arrangements.

Explicitly enumerates nested regions, matching known formulas.

Abstract

Bernardi has given a general formula for the number of regions of a deformation of the braid arrangement as a signed sum over boxed trees. We prove that each set of boxed trees which share an underlying (rooted labeled plane) tree contributes 0, +1, or -1 to this sum, and we give an algorithm for computing this value. For Ish-type arrangements, we further construct a sign-reversing involution which reduces Bernardi's signed sum to the enumeration of a set of (rooted labeled plane) trees. We conclude by explicitly enumerating the trees corresponding to the regions of Ish-type arrangements which are nested, recovering their known counting formula.