2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78508We derive a formula for expressing free cumulants whose entries are products of random variables in terms of the lattice structure of non-crossing partitions. We show the usefulness of that result by giving direct and conceptually simple proofs for a lot of results about $R$-diagonal elements. Our investigations do not assume the trace property for the considered linear functionals.26 pages, Latex2eCombinatoricsOperator AlgebrasProbabilitytext