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We study the decomposition of free random variables in terms of theirorthogonal replicas from a new perspective. First, we show that the mixedmoments of orthogonal replicas with respect to the normalized linear functional$\Phi$ are naturally described in terms of Motzkin paths identified withreduced Motzkin words. Using this fact, we demonstrate that the mixed momentsof free random variables with respect to the free product of normalized linearfunctionals are sums of the mixed moments of order $n$ of the orthogonalreplicas of these variables with respect to $\Phi$ with summation extendingover the set of reduced Motzkin words of lenght $n$. One of the applications ofthis formula is a decomposition formula for mixed moments of free randomvariables in terms of their boolean cumulants which corresponds to thedecomposition of the lattice ${\rm NC}(n)$ into sublattices $\mathcal{M}(w)$ ofpartitions which are monotonically adapted to colors in the word $w$. Thelinear functionals defined by the mixed moments of orthogonal replicas andindexed by reduced Motzkin words play the role of a generating set of the spaceof product functionals in which the boolean product corresponds to constantMotzkin paths and the free product corresponds to all Motzkin paths.

Motzkin path decompositions of functionals in noncommutative probability