A242524 Number of cyclic arrangements of S={1,2,...,n} such that the difference between any two neighbors is at least 4.

0, 0, 0, 0, 0, 0, 0, 0, 1, 24, 504, 8320, 131384, 2070087, 33465414, 561681192, 9842378284, 180447203232, 3462736479324, 69517900171056, 1458720714556848, 31955023452174314, 729874911380470641, 17359562438053760533, 429391730229931885360

Offset

1,10

Comments

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

Links

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..27 (terms a(1)-a(16) from Stanislav Sykora)

S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014.

Example

The shortest such cycle has length n=9: {1,5,9,4,8,3,7,2,6}.

Mathematica

A242524[n_] := Count[Map[lpf, Map[j1f, Permutations[Range[2, n]]]], 0]/2;
j1f[x_] := Join[{1}, x, {1}];
lpf[x_] := Length[Select[Abs[Differences[x]], # < 4 &]];
Table[A242524[n], {n, 1, 10}]
 (* OR, a less simple, but more efficient implementation. *)
A242524[n_, perm_, remain_] := Module[{opt, lr, i, new},
   If[remain == {},
     If[Abs[First[perm] - Last[perm]] >= 4, ct++];
     Return[ct],
     opt = remain; lr = Length[remain];
     For[i = 1, i <= lr, i++,
      new = First[opt]; opt = Rest[opt];
      If[Abs[Last[perm] - new] < 4, Continue[]];
      A242524[n, Join[perm, {new}],
       Complement[Range[2, n], perm, {new}]];
      ];
     Return[ct];
     ];
   ];
Table[ct = 0; A242524[n, {1}, Range[2, n]]/2, {n, 1, 12}] (* Robert Price, Oct 24 2018 *)

Prog

(C++) See the link.

Crossrefs

Cf. A242519, A242520, A242521, A242522, A242523, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242533, A242534.

Sequence in context: A112390 A166757 A289833 * A264956 A055007 A014907

Adjacent sequences: A242521 A242522 A242523 * A242525 A242526 A242527

Keyword

nonn,hard

Author

Stanislav Sykora, May 27 2014

Extensions

a(17)-a(25) from Hiroaki Yamanouchi, Aug 29 2014

Status

approved