Free cumulants and enumeration of connected partitions

Franz Lehner

Published 2001-10-02Version 1

A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected set partitions respectively. The proof relies on Moebius inversion on the partition lattice.