Let $(\mathcal A,\varphi)$ be a noncommutative probability space, let $n \in \mathbb N$, and let $a_1,\dots,a_n \in \mathcal A$. For $\pi \in NC(n)$, let $\varphi_\pi[a_1,\dots,a_n]$ denote the multiplicative moment functional over the blocks of $\pi$, and let $\kappa_\pi[a_1,\dots,a_n]$ denote the corresponding multiplicative extension of the free cumulants. If $\mu_{NC}$ denotes the Möbius function of the finite noncrossing partition lattice $NC(n)$, then for every $\sigma \in NC(n)$,

\begin{align*} \kappa_\sigma[a_1,\dots,a_n] = \sum_{\pi \in NC(n):\, \pi \leq \sigma} \mu_{NC}(\pi,\sigma)\,\varphi_\pi[a_1,\dots,a_n]. \end{align*}

In particular, for the one-block partition $1_n \in NC(n)$,

\begin{align*} \kappa_n(a_1,\dots,a_n) = \sum_{\pi \in NC(n)} \mu_{NC}(\pi,1_n)\,\varphi_\pi[a_1,\dots,a_n]. \end{align*}