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Number of aperiodic permutations of {1..n}.
7

%I #12 Aug 19 2019 16:26:49

%S 1,0,3,16,115,660,5033,39936,362718,3624920,39916789,478953648,

%T 6227020787,87177645996,1307674338105,20922779566080,355687428095983,

%U 6402373519409856,121645100408831981,2432902004460734000,51090942171698415483,1124000727695858073380

%N Number of aperiodic permutations of {1..n}.

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

%H Andrew Howroyd, <a href="/A324514/b324514.txt">Table of n, a(n) for n = 1..200</a>

%H Gus Wiseman, <a href="/A324514/a324514.png">Cycle decompositions of the a(4) = 16 aperiodic permutations</a>.

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

%F a(n) = Sum_{d|n} mu(n/d)*(n/d)^d*d!. - _Andrew Howroyd_, Aug 19 2019

%e The a(4) = 16 aperiodic permutations:

%e (1243) (1324) (1342) (1423)

%e (2134) (2314) (2413) (2431)

%e (3124) (3142) (3241) (3421)

%e (4132) (4213) (4231) (4312)

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

%o (PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ _Andrew Howroyd_, Aug 19 2019

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

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 04 2019

%E Terms a(10) and beyond from _Andrew Howroyd_, Aug 19 2019