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
KEYWORD
nonn,hard
AUTHOR
Stanislav Sykora, May 27 2014
EXTENSIONS
a(17)-a(25) from Hiroaki Yamanouchi, Aug 29 2014
STATUS
approved