Activity from matroids to rooted trees and beyond

TL;DR

This paper develops a general activity theory linking matroid bases, hyperplane arrangements, and colored labeled rooted trees, revealing new combinatorial correspondences and counting results.

Contribution

It introduces a unified activity framework applicable to NBC sets and colored trees, extending classical matroid concepts to gainic hyperplane arrangements.

Findings

Number of bounded regions equals the count of activity 0 colored rooted trees.

Establishes correspondences between hyperplane arrangement regions and combinatorial trees.

Generalizes activity notions from matroids to broader combinatorial structures.

Abstract

The interior and exterior activities of bases of a matroid are well-known notions that for instance permit one to define the Tutte polynomial. Recently, we have discovered correspondences between the regions of gainic hyperplane arrangements and coloredlabeled rooted trees. Here we define a general activity theory that applies in particular to no-broken circuit (NBC) sets and labeled colored trees. The special case of activity \textsf{0} was our motivating case. As a consequence, in a gainic hyperplane arrangement the number of bounded regions is equal to the number of the corresponding colored labeled rooted trees of activity \textsf{0}.