Sketches, moves and partitions: counting regions of deformations of reflection arrangements

TL;DR

This paper introduces a combinatorial approach using moves and sketches to count regions of reflection arrangements and their deformations, establishing bijections with non-nesting partitions and analyzing their statistics.

Contribution

It provides a uniform explicit bijection between regions of deformed reflection arrangements and non-nesting partitions, linking combinatorial structures with characteristic polynomial coefficients.

Findings

Established a bijection between regions and non-nesting partitions

Described a statistic on partitions matching characteristic polynomial coefficients

Analyzed the threshold arrangement and its deformations

Abstract

The collection of reflecting hyperplanes of a finite Coxeter group is called a reflection arrangement and it appears in many subareas of combinatorics and representation theory. We focus on the problem of counting regions of reflection arrangements and their deformations. Inspired by the recent work of Bernardi, we show that the notion of moves and sketches can be used to provide a uniform and explicit bijection between regions of (the Catalan deformation of) a reflection arrangement and certain non-nesting partitions. We then use the exponential formula to describe a statistic on these partitions such that distribution is given by the coefficients of the characteristic polynomial. Finally, we consider a sub-arrangement of type C arrangement called the threshold arrangement and its Catalan and Shi deformations.