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A242529 Number of cyclic arrangements (up to direction) of numbers 1,2,...,n such that any two neighbors are coprime.

%I A242529 #20 Oct 25 2018 17:25:43
%S A242529 1,1,1,1,6,2,36,36,360,288,11016,3888,238464,200448,3176496,4257792,
%T A242529 402573312,139511808,18240768000,11813990400,440506183680,
%U A242529 532754620416,96429560832000,32681097216000,5244692024217600,6107246661427200,490508471914905600,468867166554931200,134183696369843404800
%N A242529 Number of cyclic arrangements (up to direction) of numbers 1,2,...,n such that any two neighbors are coprime.
%C A242529 a(n)=NPC(n;S;P) is the count of all neighbor-property cycles for a specific set S={1,2,...,n} of n elements and a specific pair-property P of "being coprime". For more details, see the link and A242519.
%H A242529 S. Sykora, <a href="http://dx.doi.org/10.3247/SL5Math14.002">On Neighbor-Property Cycles</a>, <a href="http://ebyte.it/library/Library.html#math">Stan's Library</a>, Volume V, 2014.
%H A242529 Wikipedia, <a href="http://en.wikipedia.org/wiki/Coprime_integers">Coprime integers</a>
%F A242529 For n>2, a(n) = A086595(n)/2.
%e A242529 There are 6 such cycles of length n=5: C_1={1,2,3,4,5}, C_2={1,2,3,5,4},
%e A242529 C_3={1,2,5,3,4}, C_4={1,2,5,4,3}, C_5={1,3,2,5,4}, and C_6={1,4,3,2,5}.
%e A242529 For length n=6, the count drops to just 2:
%e A242529 C_1={1,2,3,4,5,6}, C_2={1,4,3,2,5,6}.
%t A242529 A242529[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;
%t A242529 j1f[x_] := Join[{1}, x, {1}];
%t A242529 lpf[x_] := Length[Select[cpf[x], # != 1 &]];
%t A242529 cpf[x_] := Module[{i},
%t A242529    Table[GCD[x[[i]], x[[i + 1]]], {i, Length[x] - 1}]];
%t A242529 Join[{1, 1}, Table[A242529[n], {n, 3, 10}]]
%t A242529 (* OR, a less simple, but more efficient implementation. *)
%t A242529 A242529[n_, perm_, remain_] := Module[{opt, lr, i, new},
%t A242529    If[remain == {},
%t A242529      If[GCD[First[perm], Last[perm]] == 1, ct++];
%t A242529      Return[ct],
%t A242529      opt = remain; lr = Length[remain];
%t A242529      For[i = 1, i <= lr, i++,
%t A242529       new = First[opt]; opt = Rest[opt];
%t A242529       If[GCD[Last[perm], new] != 1, Continue[]];
%t A242529       A242529[n, Join[perm, {new}],
%t A242529        Complement[Range[2, n], perm, {new}]];
%t A242529       ];
%t A242529      Return[ct];
%t A242529      ];
%t A242529    ];
%t A242529 Join[{1, 1},Table[ct = 0; A242529[n, {1}, Range[2, n]]/2, {n, 3, 12}] ](* _Robert Price_, Oct 25 2018 *)
%o A242529 (C++) See the link.
%Y A242529 Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242530, A242531, A242532, A242533, A242534.
%K A242529 nonn,hard
%O A242529 1,5
%A A242529 _Stanislav Sykora_, May 30 2014
%E A242529 a(1) corrected, a(19)-a(29) added by _Max Alekseyev_, Jul 04 2014