$(n,r,k)$-data set, which is an abstract tuple involving non-negative integers that would algebraically encode a permuting $(n,r,k)$-action.

\[defn:permuting\_actions\] For integers $n\geq 1 \text{ and }r,k\geq 0$, an orientation-preserving $C_n$-action $t$ on $S_g$ is called a *permuting $(n,r,k)$-action* if

(i) there is a set ${\operatorname{{\mathbb P}}}(t) \subset S_g$ of $r$ distinguished fixed points of $t$, and

(ii) there is a set ${\operatorname{{\mathbb O}}}(t) \subset S_g$ of $k$ distinguished non-trivial orbits of $t$.

\[defn:orbit\_distribution\] Let $t$ be a permuting $(n,r,k)$-action on $S_g$.

(i) Fix a point $P \in S_g$, and consider $t_{\ast}: T_P(S_g) \to T_{t(P)}(S_g)$. By the Nielsen realisation theorem [@K1], we may change $t$ by isotopy in ${\operatorname{Mod}}(S_g)$ so that $t_{\ast}$ is an isometry. Hence, $t_{\ast}$ induces a local rotation by an angle, which we shall denote by $\theta_P(t)$. Note that if $P \in {\operatorname{{\mathbb P}}}(t)$, then $\theta_P(t) = 2\pi a/n,$ where $\gcd(a,n) = 1$.

(ii) Fix an orbit ${\operatorname{{\mathbb O}}}=\{P_1,\ldots,P_s\} \in {\operatorname{{\mathbb O}}}(t)$. If $s<n$, then $s\mid n$, and there exists a cone point in the quotient orbifold of degree $n/s$. Each $P_i$ has stabilizer generated by $t^s$ and the rotation induced by $t^s$ around each $P_i$ must be the same, since its action at one point is conjugate by 