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Number of perfect necklace compositions of n.
6

%I #11 Jun 07 2019 16:32:47

%S 1,1,2,1,1,1,3,1,1,1,1,1,5,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,1,1,11,1,1,1,

%T 1,1,1,1,1,1

%N Number of perfect necklace compositions of n.

%C A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. A circular subsequence is a sequence of consecutive terms where the last and first parts are also considered consecutive. A necklace composition of n is perfect if every positive integer from 1 to n is the sum of exactly one distinct circular subsequence.

%F For n > 1, a(n) = A325787(n) + 1.

%e The a(1) = 1 , a(2) = 1, a(3) = 2, a(7) = 3, a(13) = 5, and a(31) = 11 perfect necklace compositions (A = 10, B = 11, C = 12, D = 13, E = 14):

%e 1 11 12 124 1264 12546D

%e 111 142 1327 1274C5

%e 1111111 1462 13278A

%e 1723 13625E

%e 1111111111111 15C472

%e 17324E

%e 1A8723

%e 1D6452

%e 1E4237

%e 1E5263

%e 1111111111111111111111111111111

%t neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];

%t subalt[q_]:=Union[ReplaceList[q,{___,s__,___}:>{s}],DeleteCases[ReplaceList[q,{t___,__,u___}:>{u,t}],{}]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&Sort[Total/@subalt[#]]==Range[n]&]],{n,10}]

%Y Cf. A002033, A008965, A103300, A108917, A126796, A325676, A325679, A325685, A325780, A325782, A325786, A325787, A325789.

%K nonn,more

%O 1,3

%A _Gus Wiseman_, May 22 2019