induction on $|{\operatorname{{\mathcal{C}}}}|$ to conclude that, up to isotopy, $h({\operatorname{{\mathcal{C}}}}) = {\operatorname{{\mathcal{C}}}}$.

\[defn:rkpartition\] Let ${\operatorname{{\mathcal{C}}}}$ be a multicurve of size $m$ in $S_g$. Then for integers $r,k \geq 0$, an *$(r,k)$-partition* of ${\operatorname{{\mathcal{C}}}}$ is a partition ${\operatorname{{\mathbb P}}}_{r,k}({\operatorname{{\mathcal{C}}}}) = \{{\operatorname{{\mathcal{C}}}}_1',\ldots,{\operatorname{{\mathcal{C}}}}_r',{\operatorname{{\mathcal{C}}}}_1,\ldots,{\operatorname{{\mathcal{C}}}}_k\}$ of the set ${\operatorname{{\mathcal{C}}}}$ into subsets such that for all $i$,

(i) $|{\operatorname{{\mathcal{C}}}}_i'| =1$, $|{\operatorname{{\mathcal{C}}}}_i| > 1$, and

(ii) ${\operatorname{{\mathcal{C}}}}_i$ comprises only separating or only nonseparating curves.

Let ${\operatorname{{\mathcal{C}}}}$ be a multicurve in $S_g$. Then for integers $r,k \geq 0$, a root $h$ of $t_{{\operatorname{{\mathcal{C}}}}}$ of is said to be *$(r,k)$-permuting* if it induces an $(r,k)$-partition of ${\operatorname{{\mathcal{C}}}}$.

\[notation:rkpartition\] Let ${\operatorname{{\mathcal{C}}}}$ be a multicurve of size $m$ in $S_g$ and consider an $(r,k)$-partition ${\operatorname{{\mathbb P}}}_{r,k}({\operatorname{{\mathcal{C}}}}) = \{{\operatorname{{\mathcal{C}}}}_1',\ldots,{\operatorname{{\mathcal{C}}}}_r',{\operatorname{{\mathcal{C}}}}_1,\ldots, {\operatorname{{\mathcal{C}}}}_k\}$ as in Definition \[defn:rkpartition\].

(i) We shall denote the multiset ${\Big\{\!\big\{}
    \newcommand\rrbrace{\big\}\!\Big\}}|{\operatorname{{\mathcal{C}}}}_1|,|{\operatorname{{\mathcal{C}}}}_2|, \ldots,|{\operatorname{{\mathcal{C}}}}_k|\rrbrace$ by $S({\operatorname{{\mathbb P}}}_{r,k})$. (From here on, we shall denote a multiset using ${\Big\{\!\big\{}
    \newcommand\rrbrace{\big\}\!\Big\}}\,\,\rrbrace$).

(ii) If ${\operatorname{{\mathbb P}}}_{r,k}({\operatorname{{\mathcal{C}}}})$ is induced by an $(r,k)$-permuting root $h$ of $t_{{\operatorname{{\mathcal{C}}}}}$, then we shall denote it by ${\operatorname{{\mathcal{C}}}}_{r,k}(h)$.

Permuting Actions and Permuting Data Sets {#sec: actions_data_triples}
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In this section we shall introduce permuting $(n,r,k)$-actions, which are generalizations of the nestled $(n,\ell)$-actions from [@KR1]. We shall also introduce the notion of a permuting 