Bijections between affine hyperplane arrangements and valued graphs

TL;DR

This paper introduces bijective proofs linking affine hyperplane arrangements to valued graphs, providing new combinatorial insights into the enumeration of regions for arrangements like Shi and Catalan.

Contribution

It presents novel bijections between no-broken-circuit sets of integral gain graphs and labeled binary trees, offering new proofs for known formulas of hyperplane arrangement regions.

Findings

Established bijections for Shi and Catalan arrangements

Provided combinatorial proofs for region counts

Connected hyperplane arrangements with labeled binary trees

Abstract

We show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-broken-circuit sets of the corresponding integral gain graphs and some kinds of labelled binary trees. This leads to new bijective proofs for the Shi, Catalan, and similar hyperplane arrangements.