
COMMENTS

The n! permutations (p) of the numbers 1,2,3..n may be paired (allowing duplication) in n!^2 ways. Let p’ represent a permutation of the identity 123..n: then in p x p’ = p’’, p’ will (by x) permute p (in the same way the identity was permuted) to p’’. For example, 2143 x 4321 = 3412. Iterating, 4321 x 3412 = 2143 for a period of 3. If p = p’, this recursive process gives the Pisano periods. For most other pairings the periods are of different lengths. The sequence gives the longest period that p x p’ generates for any p of length n.


PROG

(PARI) period(a, b)=my(n=matsize(a)[2], v=vector(n), aa=vector(n, i, a[i]), bb=vector(n, i, b[i]), id, nsteps); while(id!=n, for(i=1, n, v[i]=a[b[i]]); id=sum(i=1, n, b[i]==aa[i] && v[i]==bb[i]); for(i=1, n, a[i]=b[i]; b[i]=v[i]); nsteps++); nsteps
a(n)=my(a, b, m, p); for(k=1, n!, a=numtoperm(n, k); for(l=1, n!, b=numtoperm(n, l); p=period(a, b); if(p>m, m=p))); m \\ Ralf Stephan, Aug 13 2013
