\newcommand\rrbrace{\big\}\!\Big\}}(c_1,n_1),\ldots, (c_s,n_s)\rrbrace = {\Big\{\!\big\{}
    \newcommand\rrbrace{\big\}\!\Big\}}(c_1',n_1'), \ldots (c_s',n_s')\rrbrace.$$

(ii) Two permuting $(n,r,k)$-data sets $({\operatorname{{\mathcal D}}}, {\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}})$ and $({\operatorname{{\mathcal D}}}', {\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}'})$ are said to be *equivalent* if ${\operatorname{{\mathcal D}}}$ and ${\operatorname{{\mathcal D}}}'$ are equivalent as above, and ${\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}} = {\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}'}$.

Note that equivalent data sets have the same genus.

\[thm:action\_triple\_correspondence\] Given $n\geq 1$ and $g\geq 0$, equivalence classes of permuting $(n,r,k)$-data sets of genus $g$ correspond to equivalence classes of permuting $(n,r,k)$-actions on $S_g$.

Let $t$ be a permuting $(n,r,k)$-action on $S_g$ with quotient orbifold ${\operatorname{{\mathcal O}}}$ whose underlying surface has genus $g_0$. If $t$ is a free action, then ${\operatorname{{\mathcal O}}}= S_{g_0}$, and we simply write ${\operatorname{{\mathcal D}}}= (n,g_0;)$ and ${\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}} = \{(0,1),k)\}$. If $t$ is not free, let $p_j$ be the image in ${\operatorname{{\mathcal O}}}$ of the $P_j$, for $1 \leq j\leq r$, and let $q_1,q_2,\ldots, q_s$ be the other possible cone points of ${\operatorname{{\mathcal O}}}$ as in Figure \[fig:orb\].

\[fig:orb\] $p_1$ at 810 50 $p_2$ at 785 91 $q_1$ at 686 103 $q_2$ at 645 75

Let $\alpha_i$ be the generator of the orbifold fundamental group $\pi_1^{orb}({\operatorname{{\mathcal O}}})$ that goes around the point $p_i, 1\leq i\leq r$ and let 