Antisymmetry and enumeration of LL algorithms

I have been trying to enumerate all LL algorithms, (following Bernard Helmstetter's work) and cannot seem to reduce the 62208 permutations to the 1211 he has come up with. I am unaware of whether or not he used antisymmetry for his reductions. Using GAP and U rotations and reflections I have come up with a total of 8020. Initially considering antisymmetry as well (removing order 2 permutations and the identity from consideration. Note: I did not consider antisymmetric positions with dihedral symmetry or reflections) still does not bring it close to 1211. I was using Martin Schoenert's method in GAP and a very slow algorithm I wrote to determine equivalence classes from conjugation with a group (automorphism) separately with the same results. If anyone would like to see my definitions I will post them.

Also: Does anyone know of a method to employ the inverse homomorphism in GAP on the cube without forcing GAP to consider such a large space (the WHOLE cube space?).

Thank you for your time.

Comment viewing options

Select your preferred way to display the comments and click 'Save settings' to activate your changes.

Using GAP and U rotations and

Using GAP and U rotations and reflections I have come up with a total of 8020.

When you say you did U rotations, do you mean you consider two LL perms the same if they differ by a final U turn? Doing only that is not enough. Two algs LL perms may also differ by preceding it by a U turn, and that would reduce your number by a further factor 4 approx.

You can consider two LLs equivalent if they are conjugates by a U turn (or a rotation of the whole cube around the U axis, which amounts to the same thing). Together with inversion, that should get you down to 1211 (or maybe 1212 if you count the identity).

Jaap's Puzzle Page:

Thanks for the response.A

Thanks for the response.

Actually, I should have been more precise than 'U rotations'. I meant conjugation by rotations of the final layer's face. Your original interpretation that I meant permutations that differ only by a face turn was what I was missing!

Thanks again,