OFFSET
1,4
COMMENTS
A permutation is aperiodic if every rotation of {1...n} acts on the vertices of the cycle decomposition to produce a different digraph. A permutation necklace is an equivalence class of permutations under the action of rotation of vertices in the cycle decomposition. The corresponding action on words applies m -> m + 1 for m < n and n -> 1, and rotates once to the right. For example, (24531) first becomes (35142) under the application of cyclic rotation, and then is rotated right to give (23514).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
FORMULA
a(n) = A324514(n)/n.
a(n) = (1/n)*Sum_{d|n} mu(n/d)*(n/d)^d*d!. - Andrew Howroyd, Aug 19 2019
MATHEMATICA
Table[Length[Select[Permutations[Range[n]], UnsameQ@@NestList[RotateRight[#/.k_Integer:>If[k==n, 1, k+1]]&, #, n-1]&]]/n, {n, 6}]
PROG
(PARI) a(n) = (1/n)*sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ Andrew Howroyd, Aug 19 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 04 2019
EXTENSIONS
Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019
STATUS
approved