%I #5 May 12 2019 08:21:07

%S 1,2,2,4,5,6,11,18,26,38,60,90,139,213,329,501,747,1144,1712,2548,

%T 3836,5732,8442,12654,18624

%N Number of necklace compositions of n with distinct differences.

%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.

%C The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e The a(1) = 1 through a(8) = 18 necklace compositions:

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

%e (11) (12) (13) (14) (15) (16) (17)

%e (22) (23) (24) (25) (26)

%e (112) (113) (33) (34) (35)

%e (122) (114) (115) (44)

%e (132) (124) (116)

%e (133) (125)

%e (142) (134)

%e (223) (143)

%e (1132) (152)

%e (1213) (224)

%e (233)

%e (1124)

%e (1142)

%e (1214)

%e (1322)

%e (11213)

%e (11312)

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

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Differences[#]&&neckQ[#]&]],{n,15}]

%Y Cf. A000079, A008965, A235998, A242882, A318728, A325325, A325404, A325468, A325545, A325549, A325550, A325555.

%K nonn,more

%O 1,2

%A _Gus Wiseman_, May 11 2019