[ QUOTE ]
[ QUOTE ]
You mean you want a set of operations to perform on any matchup to find all equivalent matchups?
[/ QUOTE ]
Yes.
[ QUOTE ]
The equivalent matchups are all found by permuting the suits
[/ QUOTE ]
Right.
[/ QUOTE ]
Ok, there are 24 ways to permute the 4 suits.
1 permutation does nothing.
6 switch 2 suits, e.g., [img]/images/graemlins/spade.gif[/img] <-> [img]/images/graemlins/heart.gif[/img].
3 switch 2 pairs of suits, e.g., [img]/images/graemlins/spade.gif[/img] <-> [img]/images/graemlins/heart.gif[/img], [img]/images/graemlins/diamond.gif[/img] <-> [img]/images/graemlins/club.gif[/img].
8 cycle 3 suits, leaving 1 fixed, e.g., [img]/images/graemlins/spade.gif[/img] -> [img]/images/graemlins/heart.gif[/img] -> [img]/images/graemlins/diamond.gif[/img] -> [img]/images/graemlins/spade.gif[/img].
6 cycle 4 suits, e.g., [img]/images/graemlins/spade.gif[/img] -> [img]/images/graemlins/club.gif[/img] -> [img]/images/graemlins/diamond.gif[/img] -> [img]/images/graemlins/heart.gif[/img] -> [img]/images/graemlins/spade.gif[/img].
These 5 essentially different classes of permutations are indexed by the cycle structures, the 5 partitions of 4: 1+1+1+1, 2+1+1, 2+2, 3+1, and 4. Those were the 5 cases I used in my application of Burnside's Lemma.
What more do you want than these 24 operations? Note that this set of 24 does not depend on the pairs of cards.
[ QUOTE ]
f1,1 becomes A4K2 vs. Q2J3 (suit k=1 is cycled through all suits that don't otherwise appear in the matchup.)
[ QUOTE ]
Are you making these change, based on the matchup?
[/ QUOTE ]
No.
[/ QUOTE ]
You seem to have contradicted yourself. You say the operations both don't depend on the matchup, and depend on the unused suits.
Also, you hadn't mentioned f1,1 before, only f1 and f2,k.
You still haven't been clear what you mean by "cycle through." What you have said is ambiguous, since you have not stated what is involved in the cycle, and when you have more than 2 elements in a cycle, there are multiple possible cycles.
I think you should adopt a standard notation for
permutations (see the abstract algebra section of the linked page) rather than trying to invent your own notation.