to a *permuting $(n,r,k)$-action*. These are $C_n$-actions on $S_g$ that have $r$ distinguished fixed points, and $k$ distinguished non-trivial orbits. In Section \[sec: actions\_data\_triples\], we introduce the notion of a *permuting data set*, which is a generalization of a data set from [@KR1]. We use Thurston’s orbifold theory [@T1 Chapter 13] in Theorem \[thm:action\_triple\_correspondence\] to establish a correspondence between permuting $(n,r,k)$-actions on $S_g$ and permuting data sets of genus $g$ and degree $n$. In other words, permuting data sets algebraically encode these permuting actions and contain all the relevant orbit and fixed-point information required to classify the roots that will be constructed from these actions.

Let $S_g({\operatorname{{\mathcal{C}}}})$ denote the surface obtained from $S_g$ by deleting an annular neighbourhood of ${\operatorname{{\mathcal{C}}}}$ and capping. In Section \[sec:nonsepmulcurves\], we prove that conjugacy classes of roots of Dehn twists about nonseparating multicurves correspond to a special subclass of permuting actions on $S_g({\operatorname{{\mathcal{C}}}})$. We use this to obtain the following bounds for the degree of such a root

Let ${\operatorname{{\mathcal{C}}}}$ be a nonseparating multicurve in $S_g$ of size $m$, and let $h$ be an $(r,k)$-permuting root of $t_{{\operatorname{{\mathcal{C}}}}}$ of degree $n$.

(i) If $r\geq 0$, then $$n\leq \begin{cases}
    4(g-m) + 2 &: g-m \geq 1 