Dear Aner, On 30 January 2014 19:34, Aner Ben-Efraim <aner_2...@hotmail.com> wrote: > Dear Forum, > I am interested in getting all permutations of order n (order 2 is > also good) in the group SymmetricGroup([2..m]), which keep no point in > place. > I have tried to use Filtered(SymmetricGroup([2..m]),x->Order(x)=n and > NrMovedPoints(x)=m-1), but this gets extremely slow when m is double > digit. I was wondering if there is a way to do this in reasonable time > on my PC, at least for m up to say 17, or even 33 if possible.
You probably will have to stop earlier than 33; indeed, for order 2 case you essentially ask for the list of 1-factors of the complete graph on m vertices (here m must be even). The number of these things grows quite fast, in fact it is equal to (n/2-1)!! (double factorial of n/2-1). See http://oeis.org/A001147 Going through them one by one would really take quite a long time on any "ordinary" modern computer in reasonable time for m=32 (the number would be 191898783962510625). Anyhow, to do this as fast as possible you (IMHO) should not rely on GAP, but rather write your own code (or find it somewhere). Namely, for a given partition m_1+...+m_k=m of m, with all m_j>1 and dividing n, you just have to fill in the k vectors of lengths m_1,..,m_k with numbers from 1,...,m such that in each vector you have a cyclically minimal sequence. Now, do this for all such partitions (they represent possible cyclic structures of fixed point free elements of order n). This will give you all the permutations you are after, in cyclic notation. HTH, Dmitrii _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
