Directed rooted forests in higher dimension

TL;DR

This paper generalizes the concept of rooted forests from graphs to higher-dimensional cell complexes, introducing orientations and exploring their combinatorial and homological properties.

Contribution

It extends the generating function of rooted forests to cell complexes of arbitrary dimension, defining higher-dimensional rooted forests and orientations.

Findings

Generalization of rooted forest generating functions to higher dimensions

Introduction of orientations as discrete vector fields in higher-dimensional forests

Open questions on combinatorial expressions of homological quantities

Abstract

For a graph G, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.