Setup: Let $u:\Sigma \to X$ be a holomorphic map of closed Riemann surfaces with branch points $P \subset X$. For each branch point $p \in P$, we have a partition $\Gamma_p$ of $\text{deg}(u)$ given by the set of ramification numbers $\text{ram}(u,s)$ of the points $s$ mapping to $p$. $$ \Gamma_p := \{\text{ram}(u,s) \; : \; u(s) = p\} $$ In this post, I'll refer to the set of partitions $\Gamma := \{\Gamma_p\}_{p \in P}$ as the partition data of $u$.
Now consider the special case $X = \mathbb{P}^1$ is the Riemann sphere and $P = \{0,1,\infty\} \subset \mathbb{P}^1$. Fix a degree $d$ and partition data $\Gamma$. Here is question that seems like it might be elementary to me, but perhaps it's hard.
Main Question: Is there an explicit, closed formula for the following sum? Or even a relatively simple recursive formula? $$ N(\Gamma) := \sum_u \frac{1}{|\text{Aut}(u)|} $$ Here the sum is over all isomorphic classes of maps $u:\mathbb{P}^1 \to \mathbb{P}^1$ with the specified branch and partition data, and $\text{Aut}(u)$ is the group of biholomorphisms $f:\mathbb{P}^1 \to \mathbb{P}^1$ with $u \circ f = u$.
Group Action Formulation: Using Hurwitz realizability, the Main Question is equivalent to counting tuples $\sigma$ of two elements $$\sigma_0,\sigma_1 \in \text{Sym}(d)$$ subject to a condition on the sum of the sizes of the orbits of $\sigma_0,\sigma_1$ and $\sigma_0\sigma_1$ corresponding to the condition that the cover is a sphere. I don't know if this formulation is actually useful or not.
