partitions $\pi\in P(k)$ such that

-   $\pi\leq \ker i$, i.e. if $1\leq p<q\leq k$ are in the same block of $\pi$, then $i(p)=i(q)$,

-   and $\pi$ is *$({\varepsilon},i)$-noncrossing*, i.e. if there are indices$1\leq p_1<q_1<p_2<q_2\leq k$ such that $p_1$ and $p_2$ are in a block $V_p$ of $\pi$ and $q_1$ and $q_2$ are in a block $V_q$ of $\pi$ with $V_p\neq V_q$, then ${\varepsilon}_{i(p_1)i(q_1)}=1$.

The idea is, that $NC^{\varepsilon}[i]$ contains refinements of $\ker i$ which are allowed to have crossings only if the ${\varepsilon}$-entry of the crossing is 1.

-   For ${\varepsilon}={\varepsilon}_{\operatorname{comm}}$, all kinds of crossings between blocks on different indices are allowed, but not for different blocks on the same index. Hence we have: $$NC^{{\varepsilon}_{\operatorname{comm}}}[i]=\prod_{V\in\ker i} NC(V)$$

-   For ${\varepsilon}={\varepsilon}_{\operatorname{free}}$, no crossings are allowed and hence: $$NC^{{\varepsilon}_{\operatorname{free}}}[i]=\{\pi\in NC(k)\;|\;\pi\leq\ker i\}$$

We now come to the moment-cumulant formula for ${\varepsilon}$-independence found by the first author and Wysoczanski. We only formulate it for the situation of ${\varepsilon}$-independent variables (rather than for algebras).

\[MCFormula\] Let $x_1,\ldots,x_n\in A$ be ${\varepsilon}$-independent and let $i=(i(1),\ldots, i(k))\in \{1,\ldots,n\}^k$. Then: $${\varphi}(x_{i(1)}\cdots x_{i(k)})=\sum_{\pi\in NC^{\varepsilon}[i]}\kappa_\pi(x_{i(1)},\ldots, x_{i(k)})$$ Here, $\kappa_\pi(x_{i(1)},\ldots, x_{i(k)})$ is the product of the free cumulants for each block.

The ${\varepsilon}$-sphere $S_{{ {\mathbb R} },{\varepsilon}}^{n-1}$ {#SectEpsSphere}
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The sphere in ${ 