One of the definitions of simple Hurwitz number $h_{g,\mu}$ is that it counts up to automorphisms the number of ramified coverings of $\mathbb{C}P^1$ such that covering space is a connected surface of genus $g$ and covering map has only one non-regular branchpoint with the ramification profile $\mu$.

Where can I find examples of such coverings for small $g$ and $\mu$ just not to reinvent them by myself? Maybe anyone can give me a link to any paper or even provide them here? I did some work where proved that different possible ways to calculate Hurwitz numbers give the same results and now just looking for some pretty visual material to illustrate low genus and ramification profile examples.

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    $\begingroup$ Start with rational functions of one complex variable. $\endgroup$ – Moishe Kohan Feb 14 '17 at 2:03
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    $\begingroup$ note that there is no ramified covering with only one branch point, and if you don't control the number of simple critical values then the number of such coverings will be infinite $\endgroup$ – Andrey Ryabichev Feb 26 '17 at 15:05

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