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A325787 Number of perfect strict necklace compositions of n. 4
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 (list; graph; refs; listen; history; text; internal format)
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)

LINKS

Table of n, a(n) for n=1..40.

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)

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, 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,more

AUTHOR

Gus Wiseman, May 22 2019

STATUS

approved

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Last modified July 4 18:38 EDT 2020. Contains 335448 sequences. (Running on oeis4.)