A formula for simplicial tree-numbers of matroid complexes

We give a formula for the simplicial tree-numbers of the independent set complex of a finite matroid MM as a product of eigenvalues of the total combinatorial Laplacians on this complex. Two matroid invariants emerge naturally in describing the multiplicities of these eigenvalues in the formula: one is the unsigned reduced Euler characteristic of the independent set complex and the other is the ββ-invariant of a matroid. We will demonstrate various applications of this formula including a “matroid theoretic” derivation of Kalai’s simplicial tree-numbers of a standard simplex.