  Relations between moments and cumulants play a central role in both classical
and non-commutative probability theory. The latter allows for several distinct
families of cumulants corresponding to different types of independences: free,
Boolean and monotone. Relations among those cumulants have been studied
recently. In this work we focus on the problem of expressing with a closed
formula multivariate monotone cumulants in terms of free and Boolean cumulants.
In the process we introduce various constructions and statistics on
non-crossing partitions. Our approach is based on a pre-Lie algebra structure
on cumulant functionals. Relations among cumulants are described in terms of
the pre-Lie Magnus expansion combined with results on the continuous
Baker-Campbell-Hausdorff formula due to A. Murua.
