What to call permutations up to relabeling? 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


*

*2143

*2341


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.

$S_4$ (the set of permutations of 4 elements) has:
an identity
6 single transpositions.  eg $2134$
3 double transpositions. eg $2143$
8 "3 cycles" or elements of order 3.  i.e. one number is stabilized and there are two ways to exchange the remaining 3 elements. eg $3124$
6 elements of order 4. eg $2341$
These are called conjugacy classes.
For any element in a class, conjugation with any element in the group will take you to another element in the conjugacy class.  i.e. $gAg^{-1} = A$
Sounds like you want to look up conjugacy classes of symmetric groups.
