= 1$, and

(iii) $\displaystyle \sum_{i=1}^r a_i + \sum_{j=1}^s \frac{n}{n_i}c_i \equiv 0\pmod{n}$.

The number $g$ determined by the equation $$\label{eqn:riemann_hurwitz}
\frac{2-2g}{n} = 2-2g_0 + r\left(\frac{1}{n}-1 \right) + \sum_{j=1}^s \left(\frac{1}{n_j} - 1 \right)$$ is called the *genus* of the data set.

Fix an $(n,r)$-data set ${\operatorname{{\mathcal D}}}$ of genus $g$ as above.

(i) For each $(a,b) \in \{(0,1),(c_1,n_1), \ldots, (c_s,n_s)\}$, we write $$\theta((a,b)) := \begin{cases}
    0 & \text{if }a=0, \\
    2\pi a^{-1}/b &\text{if } a\neq 0.
    \end{cases}$$

(ii) For each $p \in \{(0,1),(c_1,n_1), \ldots, (c_s,n_s)\}$, choose a non-negative integer $m_p$. Then the set ${\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}} = \{(p,m_p) : m_p > 0\}$ is called an *orbit distribution* of ${\operatorname{{\mathcal D}}}$.

(iii) Given an orbit distribution ${\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}}$ associated with an $(n,r)$-data set ${\operatorname{{\mathcal D}}}$, the pair $({\operatorname{{\mathcal D}}}, {\operatorname{{\mathbb O}}}_{{\operatorname{{\mathcal D}}}})$ is called a *permuting $(n,r,k)$-data set* of genus $g$, where $k=\sum_p m_p$.

Let $$\begin{split}
{\operatorname{{\mathcal D}}}&= (n,g_0, (a_1,a_2,\ldots, a_r); (c_1,n_1), (c_2,n_2),\ldots, (c_s,n_s)) \\
\text{ and } {\operatorname{{\mathcal D}}}' &= (n,g_0', (a_1',a_2',\ldots, a_r'); (c_1',n_1'), (c_2',n_2'),\ldots, (c_s',n_s'))
\end{split}$$ be two $(n,r)$-data sets as in Definition \[defn:data\_sets\].

(i) ${\operatorname{{\mathcal D}}}$ and ${\operatorname{{\mathcal D}}}'$ are said to be *equivalent* if $${\Big\{\!\big\{}
    \newcommand\rrbrace{\big\}\!\Big\}}a_1,a_2,\ldots, a_r\rrbrace = {\Big\{\!\big\{}
    \newcommand\rrbrace{\big\}\!\Big\}}a_1', a_2', \ldots, a_r'\rrbrace, \text{ and }$$ $${\Big\{\!\big\{}
    