A325788 Number of complete strict necklace compositions of n.

1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 4, 4, 4, 0, 20, 6, 16, 12, 10, 0, 84, 40, 74, 42, 66, 38, 22, 254, 238, 188, 356, 242, 272, 150, 148, 1140, 1058, 1208, 1546, 1288

Offset

1,6

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 (counted by A032153). 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 complete if every positive integer from 1 to n is the sum of some circular subsequence.

Links

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

Example

The a(1) = 1 through a(16) = 6 complete strict necklace compositions (empty columns not shown):
  (1)  (12)  (123)  (124)  (1234)  (1253)  (1245)  (1264)  (12345)  (12634)
             (132)  (142)  (1324)  (1325)  (1326)  (1327)  (12354)  (13624)
                           (1423)  (1352)  (1542)  (1462)  (12435)  (14263)
                           (1432)  (1523)  (1623)  (1723)  (12453)  (14326)
                                                           (12543)  (14362)
                                                           (13254)  (16234)
                                                           (13425)
                                                           (13452)
                                                           (13524)
                                                           (13542)
                                                           (14235)
                                                           (14253)
                                                           (14325)
                                                           (14523)
                                                           (14532)
                                                           (15234)
                                                           (15243)
                                                           (15324)
                                                           (15342)
                                                           (15432)

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[#]&&Union[Total/@subalt[#]]==Range[n]&]], {n, 30}]

Crossrefs

Cf. A000740, A002033, A008965, A032153, A103295, A126796, A188431, A325684, A325785, A325786, A325787, A325790, A325791.

Sequence in context: A335683 A089262 A376943 * A240509 A344901 A330619

Adjacent sequences: A325785 A325786 A325787 * A325789 A325790 A325791

Keyword

nonn,more

Author

Gus Wiseman, May 22 2019

Status

approved