%I #6 May 22 2019 21:00:16

%S 1,1,2,2,4,7,12,19,41,71,141,255,509,924,1882,3395,6838,12715,25233,

%T 47049

%N Number of complete 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 first and last parts are also considered consecutive. A necklace composition of n is complete if every positive integer from 1 to n is the sum of some circular subsequence.

%e The a(1) = 1 through a(8) = 19 necklace compositions:

%e (1) (11) (12) (112) (113) (123) (124) (1124)

%e (111) (1111) (122) (132) (142) (1133)

%e (1112) (1113) (1114) (1142)

%e (11111) (1122) (1123) (1214)

%e (1212) (1132) (1223)

%e (11112) (1213) (1322)

%e (111111) (1222) (11114)

%e (11113) (11123)

%e (11122) (11132)

%e (11212) (11213)

%e (111112) (11222)

%e (1111111) (11312)

%e (12122)

%e (111113)

%e (111122)

%e (111212)

%e (112112)

%e (1111112)

%e (11111111)

%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[#]&&Union[Total/@subalt[#]]==Range[n]&]],{n,15}]

%Y Cf. A000740, A002033, A008965, A103295, A108917, A126796, A276024, A325549, A325682, A325781, A325788, A325789, A325791.

%K nonn,more

%O 1,3

%A _Gus Wiseman_, May 22 2019