My Little Pony XLVIII: I say Ol' Chap, how about those Equestrians?

[Bryn]

If the idea is to create a crackship generator, shipping more than two ponies together needs to be possible, albeit rather less likely. Also, randomised genderbending of one or more parties.

OK! So with the possibility of m-pony shipping (e.g. [1]) the general formula for all-distinct ponies is N·(N-1)·...·(N-m+1)/(m!) so the number of available ships grows as O(N^m). (For later clarity, call the first formula S)

We could also have the generator also tells us how to lay out the relationships within an m pony ship. Figuring out how many ways it can that is much more complicated - TV Tropes illustrates the cases available for a love triangle. Simplify things and make all love requited; then fail badly at working it out yourself () and look it up at Wolfram MathWorld...

The numbers of connected labeled graphs on n-nodes are 1, 1, 4, 38, 728, 26704, ... (Sloane's A001187)

which seems right for m=1,2,3 at least. Multiply the number by S to get the number of possible generator outputs containing m ponies.

If we let the ponies be assigned to any of g genders (probably taking g=3) then we get g^m times as many outputs containing m ponies.

To take a concrete example, suppose the generator deals with only the Mane Six (so n=6) assigned to three gender identity categories (g=3) and there are no clones or other ways to ship a pony with herself. There are...

• 135 two-pony ships
• 2160 three-pony ships
• 46170 four-pony ships
• 1061424 five-pony ships
• 19467216 ships involving the entire Mane Six

Leave out gender, and we get

• 15 two-pony ships (and a fic)
• 80 three-pony ships
• 570 four-pony ships
• 4368 five-pony ships
• 26704 ships involving the entire Mane Six

I'm not actually sure why I computed this, but hopefully someone will find it interesting.