Let $(\mathcal A, \varphi)$ be a noncommutative probability space, and let $(\mathcal A_i)_{i \in I}$ be a family of unital subalgebras of $\mathcal A$ that are free with respect to $\varphi$. Let $n \in \mathbb N$, let $i_1,\dots,i_n \in I$, and let $a_j \in \mathcal A_{i_j}$ for each $j \in \{1,\dots,n\}$.
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For $\pi \in NC(n)$, write $\kappa_\pi^\varphi(a_1,\dots,a_n)$ for the partition free cumulant associated to $\varphi$, so that if the blocks of $\pi$ are $V_1,\dots,V_r$ and $V_\ell = \{v_{\ell,1} < \dots < v_{\ell,m_\ell}\}$, then