Deformations of the braid arrangement and Trees

TL;DR

This paper provides counting formulas and bijections linking deformations of the braid arrangement to decorated plane trees, offering new insights into their combinatorial structure and simplifying the understanding of their regions and polynomials.

Contribution

It introduces a unified combinatorial framework connecting deformations of the braid arrangement with decorated plane trees, including explicit formulas and bijections for transitive cases.

Findings

Number of regions expressed as signed counts of decorated plane trees

Characteristic and coboundary polynomials in terms of these trees

Bijection between regions and plane trees for transitive deformations

Abstract

We establish counting formulas and bijections for deformations of the braid arrangement. Precisely, we consider real hyperplane arrangements such that all the hyperplanes are of the form $x\_i-x\_j=s$ for some integer $s$. Classical examples include the braid, Catalan, Shi, semiorder and Linial arrangements, as well as graphical arrangements. We express the number of regions of any such arrangement as a signed count of decorated plane trees. The characteristic and coboundary polynomials of these arrangements also have simple expressions in terms of these trees. We then focus on certain "well-behaved" deformations of the braid arrangement that we call transitive. This includes the Catalan, Shi, semiorder and Linial arrangements, as well as many other arrangements appearing in the literature. For any transitive deformation of the braid arrangement we establish a simple bijection between regions of the arrangement and a set of plane trees defined by local conditions. This answers a question of Gessel.