Say I have a list that may be out of order.
For instance, the list [1,3,2,4] has 2 and 3 out of place.
This would correspond to a conjugacy class of (2), meaning only one set of 2 elements is out of order (there are six ways to have a conjugacy class of (2)).
My "operation" is selecting three elements randomly, and scrambling them randomly (3! possibilities after selection).
Is there a systematic way to determine how many ways there are to go from a given conjugacy class to all other valid conjugacy classes based on my operation, besides enumerating them all brute-force?
e.g. given the operation, there are X ways to go from conjugacy class Y to conjugacy class Z, etc
A previous question, Linear algebra to find EV of sort algorithm, provides some background on this one.