A242533 - OEIS

%I #19 Oct 25 2018 21:25:52

%S 1,1,2,36,288,3888,200448,4257792,139511808,11813990400,532754620416

%N Number of cyclic arrangements of S={1,2,...,2n} such that the difference of any two neighbors is coprime to their sum.

%C a(n)=NPC(2n;S;P) is the count of all neighbor-property cycles for a specific set S of 2n elements and a specific pair-property P. For more details, see the link and A242519.

%C Conjecture: in this case it seems that NPC(n;S;P)=0 for all odd n, so only the even ones are listed. This is definitely not the case when the property P is replaced by its negation (see A242534).

%H 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.

%e For n=4, the only cycle is {1,2,3,4}.

%e The two solutions for n=6 are: C_1={1,2,3,4,5,6} and C_2={1,4,3,2,5,6}.

%t A242533[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, 2 n]]]], 0]/2;

%t j1f[x_] := Join[{1}, x, {1}];

%t lpf[x_] := Length[Select[cpf[x], ! # &]];

%t cpf[x_] := Module[{i},

%t    Table[CoprimeQ[x[[i]] - x[[i + 1]], x[[i]] + x[[i + 1]]], {i,

%t      Length[x] - 1}]];

%t Join[{1}, Table[A242533[n], {n, 2, 5}]]

%t (* OR, a less simple, but more efficient implementation. *)

%t A242533[n_, perm_, remain_] := Module[{opt, lr, i, new},

%t    If[remain == {},

%t      If[CoprimeQ[First[perm] + Last[perm], First[perm] - Last[perm]],

%t       ct++];

%t      Return[ct],

%t      opt = remain; lr = Length[remain];

%t      For[i = 1, i <= lr, i++,

%t       new = First[opt]; opt = Rest[opt];

%t       If[! CoprimeQ[Last[perm] + new, Last[perm] - new], Continue[]];

%t       A242533[n, Join[perm, {new}],

%t        Complement[Range[2, 2 n], perm, {new}]];

%t       ];

%t      Return[ct];

%t      ];

%t    ];

%t Join[{1}, Table[ct = 0; A242533[n, {1}, Range[2, 2 n]]/2, {n, 2, 6}] ](* _Robert Price_, Oct 25 2018 *)

%o (C++) See the link.

%Y Cf. A242519, A242520, A242521, A242522, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242534.

%K nonn,hard,more

%O 1,3

%A _Stanislav Sykora_, May 30 2014

%E a(10)-a(11) from _Fausto A. C. Cariboni_, May 31 2017, Jun 01 2017