A324514 Number of aperiodic permutations of {1..n}.

1, 0, 3, 16, 115, 660, 5033, 39936, 362718, 3624920, 39916789, 478953648, 6227020787, 87177645996, 1307674338105, 20922779566080, 355687428095983, 6402373519409856, 121645100408831981, 2432902004460734000, 51090942171698415483, 1124000727695858073380

Offset: 1

Author

Gus Wiseman, Mar 04 2019

Keywords

• nonn

Comments

A permutation is defined to be aperiodic if every cyclic rotation of {1..n} acts on the cycle decomposition to produce a different digraph.

Examples

			The a(4) = 16 aperiodic permutations:
  (1243) (1324) (1342) (1423)
  (2134) (2314) (2413) (2431)
  (3124) (3142) (3241) (3421)
  (4132) (4213) (4231) (4312)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Gus Wiseman, Cycle decompositions of the a(4) = 16 aperiodic permutations.

Crossrefs

Cf. A000740, A000939, A027375, A061417, A079128, A086675, A192332, A323860, A323867, A323869.

Cf. A306669, A324461, A324462, A324512, A324513, A324514.

Programs

Mathematica

Table[Length[Select[Permutations[Range[n]],UnsameQ@@NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]&]],{n,6}]

PARI

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ Andrew Howroyd, Aug 19 2019

Formula

a(n) = A306669(n) * n.

a(n) = Sum_{d|n} mu(n/d)*(n/d)^d*d!. - Andrew Howroyd, Aug 19 2019

Extensions

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019