  We establish and explore a relationship between two approaches to
moment-cumulant relations in free probability theory: on one side the main
approach, due to Speicher, given in terms of M\"obius inversion on the lattice
of noncrossing partitions, and on the other side the more recent
non-commutative shuffle-algebra approach, where the moment-cumulant relations
take the form of certain exponential-logarithm relations. We achieve this by
exhibiting two operad structures on (noncrossing) partitions, different in
nature: one is an ordinary, non-symmetric operad whose composition law is given
by insertion into gaps between elements, the other is a coloured, symmetric
operad with composition law expressing refinement of blocks. We show that these
operad structures interact so as to make the corresponding incidence bialgebra
of the former a comodule bialgebra for the latter. Furthermore, this
interaction is compatible with the shuffle structure and thus unveils how the
two approaches are intertwined. Moreover, the constructions and results are
general enough to extend to ordinary set partitions.
