Abstract
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. 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.
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