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

摘要

We present a version of the weighted cellular matrix-tree theorem that issuitable for calculating explicit generating functions for spanning trees ofhighly structured families of simplicial and cell complexes. We apply theresult to give weighted generalizations of the tree enumeration formulas ofAdin for complete colorful complexes, and of Duval, Klivans and Martin forskeleta of hypercubes. We investigate the latter further via a logarithmicgenerating function for weighted tree enumeration, and derive anothertree-counting formula using the unsigned Euler characteristics of skeleta of ahypercube and the Crapo $\beta$-invariant of uniform matroids.