## Counting maps

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:

$2\sum _{s_{(f-3)}=2f-5}^{2f-5+2} \text{...}\sum _{s_3=7}^{s_4} \sum _{s_2=5}^{s_3} \sum _{s_1=3}^{s_2} s_1\left(s_1-1\right)\left(s_2-3\right)\left(s_3-5\right)\text{...}\left(s_{(f-3)}-((2f-5)-2)\right)$

4 faces = $2\sum _{s_1=3}^5 s_1\left(s_1-1\right)$
5 faces = $2\sum _{s_2=5}^7 \sum _{s_1=3}^{s_2} s_1\left(s_1-1\right)\left(s_2-3\right)$

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: