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%I A325682 #11 Jun 19 2021 22:29:05
%S A325682 1,2,3,4,4,6,7,9,13,12,17,21,28,26,49,46,74,68,113,107,176,144,255,
%T A325682 235,375
%N A325682 Number of necklace compositions of n such that every distinct circular subsequence has a different sum.
%C A325682 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 A325682 A circular subsequence is a sequence of consecutive terms where the first and last parts are also considered consecutive.
%e A325682 The a(1) = 1 through a(8) = 13 necklace compositions:
%e A325682   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A325682        (11)  (12)   (13)    (14)     (15)      (16)       (17)
%e A325682              (111)  (22)    (23)     (24)      (25)       (26)
%e A325682                     (1111)  (11111)  (33)      (34)       (35)
%e A325682                                      (222)     (124)      (44)
%e A325682                                      (111111)  (142)      (125)
%e A325682                                                (1111111)  (152)
%e A325682                                                           (2222)
%e A325682                                                           (11111111)
%t A325682 neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t A325682 subalt[q_]:=Union[ReplaceList[q,{___,s__,___}:>{s}],DeleteCases[ReplaceList[q,{t___,__,u___}:>{u,t}],{}]];
%t A325682 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&UnsameQ@@Total/@subalt[#]&]],{n,20}]
%Y A325682 Cf. A000079, A000740, A008965, A059966, A108917, A143823, A169942, A276024.
%Y A325682 Cf. A325676, A325680, A325685, A325687.
%K A325682 nonn,more
%O A325682 1,2
%A A325682 _Gus Wiseman_, May 13 2019
%E A325682 a(21)-a(25) from _Robert Price_, Jun 19 2021

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