%I #12 Aug 24 2020 23:06:30
%S 1,0,1,0,0,2,2,0,0,4,4,4,4,0,20,6,16,12,10,0,84,40,74,42,66,38,22,254,
%T 238,188,356,242,272,150,148,1140,1058,1208,1546,1288
%N Number of complete strict necklace compositions of n.
%C A strict necklace composition of n is a finite sequence of distinct positive integers summing to n that is lexicographically minimal among all of its cyclic rotations. In other words, it is a strict composition of n starting with its least part (counted by A032153). 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 complete if every positive integer from 1 to n is the sum of some circular subsequence.
%e The a(1) = 1 through a(16) = 6 complete strict necklace compositions (empty columns not shown):
%e (1) (12) (123) (124) (1234) (1253) (1245) (1264) (12345) (12634)
%e (132) (142) (1324) (1325) (1326) (1327) (12354) (13624)
%e (1423) (1352) (1542) (1462) (12435) (14263)
%e (1432) (1523) (1623) (1723) (12453) (14326)
%e (12543) (14362)
%e (13254) (16234)
%e (13425)
%e (13452)
%e (13524)
%e (13542)
%e (14235)
%e (14253)
%e (14325)
%e (14523)
%e (14532)
%e (15234)
%e (15243)
%e (15324)
%e (15342)
%e (15432)
%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/@Select[IntegerPartitions[n],UnsameQ@@#&],neckQ[#]&&Union[Total/@subalt[#]]==Range[n]&]],{n,30}]
%Y Cf. A000740, A002033, A008965, A032153, A103295, A126796, A188431, A325684, A325785, A325786, A325787, A325790, A325791.
%K nonn,more
%O 1,6
%A _Gus Wiseman_, May 22 2019