Activity from matroids to rooted trees and beyond

2022/09/07 by Rigoberto Flórez, Flórez, Rigoberto, David Forge +1
Computer Science · Mathematics · #05C05 #05C22 #05C30 #52C35 #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Secondary 05A19

paper · pdf · doi:10.48550/arxiv.2209.03446

openalex publication_date 2022/09/07 · openalex created_date 2022/09/10 · openalex updated_date 2026/08/01

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 \textsf0 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 \textsf0.

Related

• The Active Bijection 2.a - Decomposition of activities for matroid bases, and Tutte polynomial of a matroid in terms of beta invariants of minors
• Tutte Polynomial Activities
• The Active Bijection 2.b - Decomposition of activities for oriented matroids, and general definitions of the active bijection
• The Arboricity Polynomial
• A general notion of activity for the Tutte polynomial