Weighted Tree-Numbers of Matroid Complexes

International audienceWe give a new formula for the weighted high-dimensional tree-numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes consist of polynomials in the weights. In the formula, Crapo’s $\beta$-invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree-numbers for matroid complexes.Nous présentons une nouvelle formule pour les nombres d’arbres pondérés de grande dimension des matroïdes complexes. Cette formule est dérivée du résultat que le spectre des Laplaciens combinatoires pondérés des matrides complexes sont des polynômes à plusieurs variables. Dans la formule, le $\beta$;-invariant de Crapo apparaît comme étant le facteur clé reliant les Laplaciens combinatoires pondérés et les nombres d’arbres pondérés des matroïdes complexes

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

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.Comment: 22 pages, 2 figures. Sections 6 and 7 of previous version simplified and condensed. Final version to appear in J. Combin. Theory Ser.

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

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

Weighted Tree-Numbers of Matroid Complexes

We give a new formula for the weighted high-dimensional tree-numbers of matroid complexes. This formula is derived from our result that the spectra of the weighted combinatorial Laplacians of matroid complexes consist of polynomials in the weights. In the formula, Crapo’s $\beta$-invariant appears as the key factor relating weighted combinatorial Laplacians and weighted tree-numbers for matroid complexes