Two Bijections on NBC Subsets

TL;DR

This paper introduces two explicit bijections connecting acyclic reorientations and NBC subsets of matroids, as well as regions of hyperplane arrangements to affine NBC subsets, enhancing combinatorial understanding.

Contribution

It presents novel explicit bijections linking acyclic reorientations, NBC subsets, and hyperplane arrangement regions, providing new combinatorial insights.

Findings

Established a bijection between acyclic reorientations and NBC subsets.

Linked regions of hyperplane arrangements to affine NBC subsets.

Enhanced understanding of combinatorial structures in matroids and arrangements.

Abstract

We establish two explicit bijections: from acyclic reorientations of an oriented matroid to no broken circuit (NBC) subsets of its underlying matroid, and from regions of a real hyperplane arrangement to its affine NBC subsets.