Let $(A,\varphi)$ be a noncommutative probability space, with $A$ a unital complex algebra and $\varphi: A \to \mathbb{C}$ a unital linear functional. Let $n \in \mathbb{N}$, and let $x_1,\dots,x_n \in A$. For each $m \in \mathbb{N}$, let $\kappa_m: A^m \to \mathbb{C}$ denote the $m$-th free cumulant functional associated to $\varphi$. For $\pi \in NC(n)$, write

\begin{align*} \kappa_\pi[x_1,\dots,x_n] := \prod_{V \in \pi} \kappa_{|V|}(x_{i_1},\dots,x_{i_{|V|}}) \end{align*}

where each block $V \in \pi$ is written in increasing order as $V = \{i_1 < \cdots < i_{|V|}\}$. If $1_n \in NC(n)$ denotes the maximal one-block partition, then

\begin{align*} \kappa_n(x_1,\dots,x_n)=\varphi(x_1\cdots x_n)-\sum_{\substack{\pi\in NC(n) : \pi\ne 1_n}} \kappa_\pi[x_1,\dots,x_n]. \end{align*}

Consequently, the moment functionals

\begin{align*} M_m: A^m &\to \mathbb{C} \end{align*}

\begin{align*} (y_1,\dots,y_m) &\mapsto \varphi(y_1\cdots y_m) \end{align*}

for $1 \le m \le n$ determine the free cumulant functionals $\kappa_m$ for $1 \le m \le n$, and the free cumulant functionals $\kappa_m$ for $1 \le m \le n$ determine the moment functionals $M_m$ for $1 \le m \le n$.