A325786 Number of complete necklace compositions of n.

1, 1, 2, 2, 4, 7, 12, 19, 41, 71, 141, 255, 509, 924, 1882, 3395, 6838, 12715, 25233, 47049

Offset: 1

Author

Gus Wiseman, May 22 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. A necklace composition of n is complete if every positive integer from 1 to n is the sum of some circular subsequence.

Examples

			The a(1) = 1 through a(8) = 19 necklace compositions:
  (1)  (11)  (12)   (112)   (113)    (123)     (124)      (1124)
             (111)  (1111)  (122)    (132)     (142)      (1133)
                            (1112)   (1113)    (1114)     (1142)
                            (11111)  (1122)    (1123)     (1214)
                                     (1212)    (1132)     (1223)
                                     (11112)   (1213)     (1322)
                                     (111111)  (1222)     (11114)
                                               (11113)    (11123)
                                               (11122)    (11132)
                                               (11212)    (11213)
                                               (111112)   (11222)
                                               (1111111)  (11312)
                                                          (12122)
                                                          (111113)
                                                          (111122)
                                                          (111212)
                                                          (112112)
                                                          (1111112)
                                                          (11111111)
		

Crossrefs

Cf. A000740, A002033, A008965, A103295, A108917, A126796, A276024, A325549, A325682, A325781, A325788, A325789, A325791.

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[#]&&Union[Total/@subalt[#]]==Range[n]&]],{n,15}]