Garrett Figueroa on Nostr: Here's a cute combinatorial conundrum. I'm certain this has been thought about ...

Here's a cute combinatorial conundrum. I'm certain this has been thought about before, but this is my first time thinking about it and I'm not sure if this problem has a standard name.

Let [𝑛] be the set with 𝑛∈ℕ elements, let 𝑆ₙ↻[𝑛] be the usual action by permutations, and uniformly pick σ∈𝑆ₙ. Now restrict this 𝑆ₙ-action to <σ>, the subgroup generated by σ, and let 𝑓^σ=<σ>↻[𝑛] .

1. What is the expected value of the number of orbits of 𝑓^σ?

2. Now make sure σ has no 1-cycles, i.e. we uniformly pick σ from the subset of 𝑆ₙ of permutations without 1-cycles. Now what is the expected value of the number of orbits of 𝑓^σ?

This is straightforward to compute for small 𝑛 - off the dome, here are the first few for the full 𝑆ₙ from 𝑛=0:

0, 1, 1.5, 11/6, ...

And for 𝑆ₙ less permutations with 1-cycles, 𝑛=2 gives 1 orbit and 𝑛=3 gives (1+1+3)/3=5/3 orbits.

I'm sure this is interesting! And may toy around with some computations eventually, get some practice with Python and so on, but figured I'd ask y'all first if this has been thought about before.

#math #puzzles