A325787 Number of perfect strict necklace compositions of n.

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, 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, 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

Offset

1,7

Comments

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:

1: (1)

2: (2)

3: (1,2)

4: (4)

5: (4,1)

6: (6)

7: (4,1,2)

8: (2,6)

9: (1,2,6)

10: (6,4)

11: (6,4,1)

12: (2,6,4)

13: (1,2,6,4)

a(n) > 0 iff n = A002061(k) = A004136(k) for some k. - Bert Dobbelaere, Nov 11 2020

Links

Bert Dobbelaere, Table of n, a(n) for n = 1..306

Example

The a(1) = 1 through a(31) = 10 perfect strict necklace compositions (empty columns not shown):
  (1)  (1,2)  (1,2,4)  (1,2,6,4)  (1,3,10,2,5)  (1,10,8,7,2,3)
              (1,4,2)  (1,3,2,7)  (1,5,2,10,3)  (1,13,6,4,5,2)
                       (1,4,6,2)                (1,14,4,2,3,7)
                       (1,7,2,3)                (1,14,5,2,6,3)
                                                (1,2,5,4,6,13)
                                                (1,2,7,4,12,5)
                                                (1,3,2,7,8,10)
                                                (1,3,6,2,5,14)
                                                (1,5,12,4,7,2)
                                                (1,7,3,2,4,14)
From Bert Dobbelaere, Nov 11 2020: (Start)
Compositions matching nonzero terms from a(57) to a(273), up to symmetry.
a(57) = 12:
  (1,2,10,19,4,7,9,5)
  (1,3,5,11,2,12,17,6)
  (1,3,8,2,16,7,15,5)
  (1,4,2,10,18,3,11,8)
  (1,4,22,7,3,6,2,12)
  (1,6,12,4,21,3,2,8)
a(73) = 8:
  (1,2,4,8,16,5,18,9,10)
  (1,4,7,6,3,28,2,8,14)
  (1,6,4,24,13,3,2,12,8)
  (1,11,8,6,4,3,2,22,16)
a(91) = 12:
  (1,2,6,18,22,7,5,16,4,10)
  (1,3,9,11,6,8,2,5,28,18)
  (1,4,2,20,8,9,23,10,3,11)
  (1,4,3,10,2,9,14,16,6,26)
  (1,5,4,13,3,8,7,12,2,36)
  (1,6,9,11,29,4,8,2,3,18)
a(133) = 36:
  (1,2,9,8,14,4,43,7,6,10,5,24)
  (1,2,12,31,25,4,9,10,7,11,16,5)
  (1,2,14,4,37,7,8,27,5,6,13,9)
  (1,2,14,12,32,19,6,5,4,18,13,7)
  (1,3,8,9,5,19,23,16,13,2,28,6)
  (1,3,12,34,21,2,8,9,5,6,7,25)
  (1,3,23,24,6,22,10,11,18,2,5,8)
  (1,4,7,3,16,2,6,17,20,9,13,35)
  (1,4,16,3,15,10,12,14,17,33,2,6)
  (1,4,19,20,27,3,6,25,7,8,2,11)
  (1,4,20,3,40,10,9,2,15,16,6,7)
  (1,5,12,21,29,11,3,16,4,22,2,7)
  (1,7,13,12,3,11,5,18,4,2,48,9)
  (1,8,10,5,7,21,4,2,11,3,26,35)
  (1,14,3,2,4,7,21,8,25,10,12,26)
  (1,14,10,20,7,6,3,2,17,4,8,41)
  (1,15,5,3,25,2,7,4,6,12,14,39)
  (1,22,14,20,5,13,8,3,4,2,10,31)
a(183) = 40:
  (1,2,13,7,5,14,34,6,4,33,18,17,21,8)
  (1,2,21,17,11,5,9,4,26,6,47,15,12,7)
  (1,2,28,14,5,6,9,12,48,18,4,13,16,7)
  (1,3,5,6,25,32,23,10,18,2,17,7,22,12)
  (1,3,12,7,20,14,44,6,5,24,2,28,8,9)
  (1,3,24,6,12,14,11,55,7,2,8,5,16,19)
  (1,4,6,31,3,13,2,7,14,12,17,46,8,19)
  (1,4,8,52,3,25,18,2,9,24,6,10,7,14)
  (1,4,20,2,12,3,6,7,33,11,8,10,35,31)
  (1,5,2,24,15,29,14,21,13,4,33,3,9,10)
  (1,5,23,27,42,3,4,11,2,19,12,10,16,8)
  (1,6,8,22,4,5,33,21,3,20,32,16,2,10)
  (1,8,3,10,23,5,56,4,2,14,15,17,7,18)
  (1,8,21,45,6,7,11,17,3,2,10,4,23,25)
  (1,9,5,40,3,4,21,35,16,18,2,6,11,12)
  (1,9,14,26,4,2,11,5,3,12,27,34,7,28)
  (1,9,21,25,3,4,8,5,6,16,2,36,14,33)
  (1,10,22,34,27,12,3,4,2,14,24,5,8,17)
  (1,10,48,9,19,4,8,6,7,17,3,2,34,15)
  (1,12,48,6,2,38,3,22,7,10,11,5,4,14)
a(273) = 12:
  (1,2,4,8,16,32,27,26,11,9,45,13,10,29,5,17,18)
  (1,3,12,10,31,7,27,2,6,5,19,20,62,14,9,28,17)
  (1,7,3,15,33,5,24,68,2,14,6,17,4,9,19,12,34)
  (1,7,12,44,25,41,9,17,4,6,22,33,13,2,3,11,23)
  (1,7,31,2,11,3,9,36,17,4,22,6,18,72,5,10,19)
  (1,21,11,50,39,13,6,4,14,16,25,26,3,2,7,8,27)
(End)

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/@Select[IntegerPartitions[n], UnsameQ@@#&], neckQ[#]&&Sort[Total/@subalt[#]]==Range[n]&]], {n, 30}]

Crossrefs

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

Sequence in context: A282568 A028833 A024943 * A005929 A005871 A005888

Adjacent sequences: A325784 A325785 A325786 * A325788 A325789 A325790

Keyword

nonn

Author

Gus Wiseman, May 22 2019

Extensions

More terms from Bert Dobbelaere, Nov 11 2020

Status

approved