Let $(\mathcal A,\varphi)$ be a complex noncommutative probability space, where $\mathcal A$ is a unital algebra over $\mathbb C$ and $\varphi:\mathcal A\to\mathbb C$ is a unital linear functional. Let $(\mathcal A_i)_{i\in I}$ be a family of free unital subalgebras of $\mathcal A$. Let $n\in\mathbb N$, let $i_1,\dots,i_n\in I$, and let $a_j\in\mathcal A_{i_j}$ for each $j\in\{1,\dots,n\}$. For each $m\in\mathbb N$, let $\kappa_m:\mathcal A^m\to\mathbb C$ denote the $m$-th free cumulant associated to $\varphi$. For each $i\in I$, let $\varphi_i:\mathcal A_i\to\mathbb C$ be the restriction of $\varphi$ to $\mathcal A_i$, and let $\kappa_m^i:\mathcal A_i^m\to\mathbb C$ denote the $m$-th free cumulant associated to the marginal noncommutative probability space $(\mathcal A_i,\varphi_i)$.

Define the colour map $c:\{1,\dots,n\}\to I$ by $c(j)=i_j$. Let $NC(n)$ be the finite set of noncrossing partitions of $\{1,\dots,n\}$, and let $NC_c(n)$ be the subset of those partitions whose every block has constant colour. If $B=\{b_1<\cdots<b_m\}\subseteq\{1,\dots,n\}$ has constant colour, let $i(B)\in I$ be the unique index satisfying $i_{b_r}=i(B)$ for every $r\in\{1,\dots,m\}$, and define $a_B=(a_{b_1},\dots,a_{b_m})\in\mathcal A_{i(B)}^m$. Then

\begin{align*} \varphi(a_1\cdots a_n)=\sum_{\pi\in NC_c(n)}\prod_{B\in\pi}\kappa_{|B|}^{i(B)}(a_B). \end{align*}

Consequently, the mixed moment $\varphi(a_1\cdots a_n)$ is a finite polynomial, with integer coefficients determined only by the colour pattern $(i_1,\dots,i_n)$, in the marginal free cumulants of tuples drawn from the individual subalgebras $\mathcal A_i$.