login
Number of complete strict necklace compositions of n.
8

%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