Thanks for asking! (but I bet you're sorry you did)

You're not wrong there :) :)

Its situations like this where MathML or equivalent would be really nice, since you could just give me the forumlae :)

In essence, what I understood was the following:

For every permutation 1->n check every permutation 1->n to see if it is equivalent according to the rules. If it is, mark it as such.

Results: a set of groups of permutations who are deemed equivalent.

Are the rules transitive?

My head generated psuedocode something like this (no efficiency or anything, just a general algorithm):

foreach $p_1 (permutation(1..n)) {
   if ($tags{$p_1}) { next; }
   $tags{$p_1} = $p_1;
   foreach $p_2 (permutation(1..n)) {
       if (rule1($p_1, $p_2) && rule2($p_1,$p_2) && rule3($p_1, $p_2))
+ {
           $tags{$p_2} = $p_1;
       }
   }
}
[download]

But that only works if the rules are both transitive and symmetric (you hint that they are but I may be reading it wrong)

I would be interested in the contents of the rules 1->3 if they're not going to make my head ache :)