I’ve posted this question on mathoverflow.

Is there a formula to count how many different topological regular maps can be created with n faces (on a sphere)?

- For “regular” I intend maps in which the boundaries form a 3-regular planar graph
- For “different” I intend maps that cannot be topologically transformed one into another (faces have to be considered unnamed)

I’ve been looking for a formula, but it is too difficult for me. Maybe it has a simple solution but I don’t see it.

This was my best guess, but I already know that it is not correct because full of symmetries, as it can be verified manually.

General formula:

4 faces =

5 faces =

Here are the first results that can be found manually (excluding simmetrical maps):

- 2 faces = 0 possible regular map (an island and the ocean) (not to be counted, because not regular)
- 3 faces = 1 possible regular map (an island with two regions and the ocean) (two islands and the ocean wouldn’t be regular)
- 4 faces = 3 possible regular maps (can be verified adding a face from the previous map)
- 5 faces = 16 possible regular maps (with homeomorphics eliminated)
- …

These are all maps up to 5 faces:

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