A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes

TL;DR

This paper develops a weighted cellular matrix-tree theorem applicable to complex structures like simplicial and cell complexes, enabling explicit enumeration of spanning trees with applications to colorful complexes and hypercube skeleta.

Contribution

It introduces a new weighted matrix-tree theorem for cellular complexes and applies it to generalize existing tree enumeration formulas for specific complex families.

Findings

Derived explicit generating functions for spanning trees in structured complexes

Generalized tree enumeration formulas for colorful complexes and hypercube skeleta

Connected tree counts to Euler characteristics and matroid invariants

Abstract

We present a version of the weighted cellular matrix-tree theorem that is suitable for calculating explicit generating functions for spanning trees of highly structured families of simplicial and cell complexes. We apply the result to give weighted generalizations of the tree enumeration formulas of Adin for complete colorful complexes, and of Duval, Klivans and Martin for skeleta of hypercubes. We investigate the latter further via a logarithmic generating function for weighted tree enumeration, and derive another tree-counting formula using the unsigned Euler characteristics of skeleta of a hypercube and the Crapo $\beta$-invariant of uniform matroids.