%I #25 Jan 08 2021 22:02:47

%S 1,0,1,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,10,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Number of perfect 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. 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. For example, the composition (1,2,6,4) is perfect because it has the following circular subsequences and sums:

%C 1: (1)

%C 2: (2)

%C 3: (1,2)

%C 4: (4)

%C 5: (4,1)

%C 6: (6)

%C 7: (4,1,2)

%C 8: (2,6)

%C 9: (1,2,6)

%C 10: (6,4)

%C 11: (6,4,1)

%C 12: (2,6,4)

%C 13: (1,2,6,4)

%C a(n) > 0 iff n = A002061(k) = A004136(k) for some k. - _Bert Dobbelaere_, Nov 11 2020

%H Bert Dobbelaere, <a href="/A325787/b325787.txt">Table of n, a(n) for n = 1..306</a>

%e The a(1) = 1 through a(31) = 10 perfect strict necklace compositions (empty columns not shown):

%e (1) (1,2) (1,2,4) (1,2,6,4) (1,3,10,2,5) (1,10,8,7,2,3)

%e (1,4,2) (1,3,2,7) (1,5,2,10,3) (1,13,6,4,5,2)

%e (1,4,6,2) (1,14,4,2,3,7)

%e (1,7,2,3) (1,14,5,2,6,3)

%e (1,2,5,4,6,13)

%e (1,2,7,4,12,5)

%e (1,3,2,7,8,10)

%e (1,3,6,2,5,14)

%e (1,5,12,4,7,2)

%e (1,7,3,2,4,14)

%e From _Bert Dobbelaere_, Nov 11 2020: (Start)

%e Compositions matching nonzero terms from a(57) to a(273), up to symmetry.

%e a(57) = 12:

%e (1,2,10,19,4,7,9,5)

%e (1,3,5,11,2,12,17,6)

%e (1,3,8,2,16,7,15,5)

%e (1,4,2,10,18,3,11,8)

%e (1,4,22,7,3,6,2,12)

%e (1,6,12,4,21,3,2,8)

%e a(73) = 8:

%e (1,2,4,8,16,5,18,9,10)

%e (1,4,7,6,3,28,2,8,14)

%e (1,6,4,24,13,3,2,12,8)

%e (1,11,8,6,4,3,2,22,16)

%e a(91) = 12:

%e (1,2,6,18,22,7,5,16,4,10)

%e (1,3,9,11,6,8,2,5,28,18)

%e (1,4,2,20,8,9,23,10,3,11)

%e (1,4,3,10,2,9,14,16,6,26)

%e (1,5,4,13,3,8,7,12,2,36)

%e (1,6,9,11,29,4,8,2,3,18)

%e a(133) = 36:

%e (1,2,9,8,14,4,43,7,6,10,5,24)

%e (1,2,12,31,25,4,9,10,7,11,16,5)

%e (1,2,14,4,37,7,8,27,5,6,13,9)

%e (1,2,14,12,32,19,6,5,4,18,13,7)

%e (1,3,8,9,5,19,23,16,13,2,28,6)

%e (1,3,12,34,21,2,8,9,5,6,7,25)

%e (1,3,23,24,6,22,10,11,18,2,5,8)

%e (1,4,7,3,16,2,6,17,20,9,13,35)

%e (1,4,16,3,15,10,12,14,17,33,2,6)

%e (1,4,19,20,27,3,6,25,7,8,2,11)

%e (1,4,20,3,40,10,9,2,15,16,6,7)

%e (1,5,12,21,29,11,3,16,4,22,2,7)

%e (1,7,13,12,3,11,5,18,4,2,48,9)

%e (1,8,10,5,7,21,4,2,11,3,26,35)

%e (1,14,3,2,4,7,21,8,25,10,12,26)

%e (1,14,10,20,7,6,3,2,17,4,8,41)

%e (1,15,5,3,25,2,7,4,6,12,14,39)

%e (1,22,14,20,5,13,8,3,4,2,10,31)

%e a(183) = 40:

%e (1,2,13,7,5,14,34,6,4,33,18,17,21,8)

%e (1,2,21,17,11,5,9,4,26,6,47,15,12,7)

%e (1,2,28,14,5,6,9,12,48,18,4,13,16,7)

%e (1,3,5,6,25,32,23,10,18,2,17,7,22,12)

%e (1,3,12,7,20,14,44,6,5,24,2,28,8,9)

%e (1,3,24,6,12,14,11,55,7,2,8,5,16,19)

%e (1,4,6,31,3,13,2,7,14,12,17,46,8,19)

%e (1,4,8,52,3,25,18,2,9,24,6,10,7,14)

%e (1,4,20,2,12,3,6,7,33,11,8,10,35,31)

%e (1,5,2,24,15,29,14,21,13,4,33,3,9,10)

%e (1,5,23,27,42,3,4,11,2,19,12,10,16,8)

%e (1,6,8,22,4,5,33,21,3,20,32,16,2,10)

%e (1,8,3,10,23,5,56,4,2,14,15,17,7,18)

%e (1,8,21,45,6,7,11,17,3,2,10,4,23,25)

%e (1,9,5,40,3,4,21,35,16,18,2,6,11,12)

%e (1,9,14,26,4,2,11,5,3,12,27,34,7,28)

%e (1,9,21,25,3,4,8,5,6,16,2,36,14,33)

%e (1,10,22,34,27,12,3,4,2,14,24,5,8,17)

%e (1,10,48,9,19,4,8,6,7,17,3,2,34,15)

%e (1,12,48,6,2,38,3,22,7,10,11,5,4,14)

%e a(273) = 12:

%e (1,2,4,8,16,32,27,26,11,9,45,13,10,29,5,17,18)

%e (1,3,12,10,31,7,27,2,6,5,19,20,62,14,9,28,17)

%e (1,7,3,15,33,5,24,68,2,14,6,17,4,9,19,12,34)

%e (1,7,12,44,25,41,9,17,4,6,22,33,13,2,3,11,23)

%e (1,7,31,2,11,3,9,36,17,4,22,6,18,72,5,10,19)

%e (1,21,11,50,39,13,6,4,14,16,25,26,3,2,7,8,27)

%e (End)

%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[#]&&Sort[Total/@subalt[#]]==Range[n]&]],{n,30}]

%Y Cf. A002033, A002061, A004136, A008965, A032153, A103300, A126796, A325680, A325682, A325780, A325782, A325786, A325788, A325789.

%K nonn

%O 1,7

%A _Gus Wiseman_, May 22 2019

%E More terms from _Bert Dobbelaere_, Nov 11 2020