I want to know about the properties of permutations when you ignore the specific elements that were permuted and instead look out how they were permuted.
For example, with permutations of the digits 1, 2, 3 we have six
- 123 A
- 132 B
- 213 B
- 231 C
- 312 C
- 321 B
But I want to group them in equivalence classes according to the letters. A is the identity, B is a single swap, and C is a rotation. In this case you can look at the number of fixed points to get the equivalence classes, but for larger numbers that doesn't work. For example for four we have two classes with no fixed points
Another way to think of it is as digraphs where every node has indegree=outdegree=1 and graphs are equivalent if they are isomorphic.
Is there a name for this kind of permutation? I'm having great difficulty looking it up.