Let $(\mathcal A,\varphi)$ be a noncommutative probability space, where $\mathcal A$ is a unital complex algebra and $\varphi:\mathcal A\to\mathbb C$ is a unital linear functional. For every integer $n\geq 1$, let $NC(n)$ denote the finite lattice of noncrossing partitions of $\{1,\dots,n\}$ ordered by refinement. Then there exists a unique family of multilinear maps $\kappa_n:\mathcal A^n\to\mathbb C$, for $n\geq 1$, such that for every $n\geq 1$ and every $a_1,\dots,a_n\in\mathcal A$,

\begin{align*} \varphi(a_1\cdots a_n)=\sum_{\pi\in NC(n)}\kappa_\pi[a_1,\dots,a_n]. \end{align*}

Here, if $V=\{i_1<\cdots<i_r\}$ is a block of $\pi$, then $\kappa_V[a_1,\dots,a_n]:=\kappa_r(a_{i_1},\dots,a_{i_r})$, and

\begin{align*} \kappa_\pi[a_1,\dots,a_n]:=\prod_{V\in\pi}\kappa_V[a_1,\dots,a_n]. \end{align*}

The product is a finite product in the commutative field $\mathbb C$, so it is independent of the order in which the blocks are listed. The maps $\kappa_n$ are called the free cumulants of $(\mathcal A,\varphi)$.