OFFSET
1,2
COMMENTS
The n! permutations (p) of the numbers 1,2,3..n may be paired (allowing duplication) in n!^2 ways. For a pair of permutations (p, p'), let p'' = p x p', p''' = p' x p'', and so on until the starting pair (p, p') is obtained. If p = p', this iterative process gives the Pisano periods. For most other pairs the periods have different lengths. The sequence gives the longest period that (p, p') generates for any p, p' of length n.
Period is invariant with respect to simultaneous conjugation of both p, p'. - Max Alekseyev, Feb 09 2024
LINKS
EXAMPLE
For n = 4: 3142 x 2341 = 1423; 2341 x 1423 = 2134... the sequence thus generated is of period = 18.
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
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Russell Walsmith, Jun 13 2013
EXTENSIONS
a(6) from Ralf Stephan, Aug 13 2013
Edited and a(7)-a(11) added by Max Alekseyev, Feb 13 2024
STATUS
approved