A conceptual proof of a basic result in the combinatorial approach to freeness

We present a conceptual proof for the theorem describing free cumulants wit h products as arguments. We reduce that theorem to a statement about the be havior of Mobius inversion under the embedding of lattices. As direct conse quence of this insight we get generalizations of the theorem to the operato r-valued case as well as analogous statements fur classical and Boolean cum ulants.