Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

We consider the group $(\mathcal{G},*)$ of unitized multiplicative functions in incidence algebra non-crossing partitions, where ``$*$'' denotes convolution operation. introduce a larger $(\widetilde{\mathcal{G}},*)$ from same algebra, which satisfy weaker condition being ``semi-multiplicative''. The natural action $\widetilde{\mathcal{G}}$ on sequences multilinear functionals non-commutative probability space captures combinatorics transitions between moments and some brands cumulants that are studied literature. use framework order to explain why multiplication free random variables can be very nicely described terms Boolean more generally $t$-Boolean cumulants, one-parameter interpolation arising work Bozejko Wysoczanski. It is known $\mathcal{G}$ naturally identified as characters Hopf Sym symmetric functions. show also $\mathcal{T}$, an sense Schmitt. Moreover, inclusion into turns out dual bialgebra homomorphism $\mathcal{T}$ onto Sym.