  Free cumulants are multilinear functionals defined in terms of the moment
functional with the use of the family of lattices of noncrossing partitions. In
the univariate case, they can be identified with the coefficients of the
Voiculescu transform of the moment functional which plays a role similar to
that of the logarithm of the Fourier transform. The associated linearization
property is connected with free independence. In turn, the family of much
smaller lattices of interval partitions is used to define Boolean cumulants
connected with Boolean independence. In order to bridge the gap between these
two families of lattices and the associated cumulants we introduce and study
the family of lattices of noncrossing partitions adapted to Motzkin paths and
define the associated operator-valued `Motzkin cumulants'. We prove the
corresponding M\"{o}bius inversion formula which plays the role of a lattice
refinement of the formula expressing free cumulants in terms of Boolean
cumulants. We apply this concept to free probability and obtain the additive
decomposition of free cumulants in terms of scalar-valued counterparts of
Motzkin cumulants.
