A325682 Number of necklace compositions of n such that every distinct circular subsequence has a different sum.

1, 2, 3, 4, 4, 6, 7, 9, 13, 12, 17, 21, 28, 26, 49, 46, 74, 68, 113, 107, 176, 144, 255, 235, 375

Offset: 1

Author

Gus Wiseman, May 13 2019

Keywords

• nonn
• more

Comments

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.

A circular subsequence is a sequence of consecutive terms where the first and last parts are also considered consecutive.

Examples

			The a(1) = 1 through a(8) = 13 necklace compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (1111)  (11111)  (33)      (34)       (35)
                                     (222)     (124)      (44)
                                     (111111)  (142)      (125)
                                               (1111111)  (152)
                                                          (2222)
                                                          (11111111)
		

Crossrefs

Cf. A000079, A000740, A008965, A059966, A108917, A143823, A169942, A276024.

Cf. A325676, A325680, A325685, A325687.

Programs

Mathematica

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
subalt[q_]:=Union[ReplaceList[q,{_,s__,_}:>{s}],DeleteCases[ReplaceList[q,{t___,,u___}:>{u,t}],{}]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&UnsameQ@@Total/@subalt[#]&]],{n,20}]

Extensions

a(21)-a(25) from Robert Price, Jun 19 2021