a power of $t$ to its action at each other point in the orbit. So the rotation angle is of the form $2\pi c^{-1}/(n/s)\pmod{2\pi}$, where $(c,n/s) = 1$ and $c^{-1}$ denotes the inverse of $c\pmod{n/s}$. We now associate to this orbit a pair $p({\operatorname{{\mathbb O}}})$ as follows: $$p({\operatorname{{\mathbb O}}}) := \begin{cases}
    (c, n/s) &\text{if } s<n \\
    (0,1) &\text{if } s=n.
    \end{cases}$$

(iii) For any orbit ${\operatorname{{\mathbb O}}}\in {\operatorname{{\mathbb O}}}(t)$, if $p({\operatorname{{\mathbb O}}}) = (a,b)$, then we define $$\theta_{{\operatorname{{\mathbb O}}}}(t) := 
    \begin{cases}
    2\pi a^{-1}/b &\text{if } a\neq 0 \\
    0 &\text{if }a=0.
    \end{cases}$$

\[defn: orbit\_distribution\] Consider a permuting $(n,r,k)$-action $t$ on $S_g$ with ${\operatorname{{\mathbb P}}}(t) = \{P_1,\ldots,P_r\}$ and ${\operatorname{{\mathbb O}}}(t) = \{{\operatorname{{\mathbb O}}}_1,{\operatorname{{\mathbb O}}}_2,\ldots, {\operatorname{{\mathbb O}}}_k\}$.

(i) We write $$S(t) = {\Big\{\!\big\{}
    \newcommand\rrbrace{\big\}\!\Big\}}|{\operatorname{{\mathbb O}}}_1|, |{\operatorname{{\mathbb O}}}_2|, \ldots, |{\operatorname{{\mathbb O}}}_k|\rrbrace.$$

(ii) For each $p \in \{p({\operatorname{{\mathbb O}}}_i) : 1\leq i \leq k\}$, define $$m_p = |\{j : p({\operatorname{{\mathbb O}}}_j) = x\}|.$$ We define the *orbit distribution* of $t$ to be the set $${\operatorname{{\mathbb O}}}_t = \{(p,m_p) :  \in \{p({\operatorname{{\mathbb O}}}_i): 1\leq i\leq k\}\}.$$

\[defn:eq\_perm\_actions\] Let $t_1$ and $t_2$ be two permuting $(n,r,k)$-actions on $S_g$ with ${\operatorname{{\mathbb P}}}(t_s) = \{P_{s,1},P_{s,2},\ldots,P_{s,r}\}$ and ${\operatorname{{\mathbb O}}}(t_s) = \{{\operatorname{{\mathbb O}}}_{s,1},{\operatorname{{\mathbb O}}}_{s,2},\ldots, {\operatorname{{\mathbb O}}}_{s,k}\}$ for $s=1,2$. We 