Classical, free and Boolean independence is characterized by the vanishing of the associated mixed cumulants.
This is not the case for monotone independence, the reason being the failure of exchangeability which is replaced by the weaker property of
spreadability. In joint work with T. Hasebe we could show that in this generalized setting mixed cumulants can be described in terms of Goldberg coefficients.
This development has a parallel in the transition from symmetric functions to quasisymmetric functions and indeed Novelli and Thibon pointed out that the cumulant identities can be interpreted as identities satisfied by Eulerian idempotents in the Hopf algebra WQSym of word quasi-symmetric functions. We report on our joint effort to understand this correspondence.