A325555 Number of necklace compositions of n with distinct differences up to sign.

1, 2, 2, 4, 5, 6, 10, 15, 19, 24, 39, 49, 78, 106, 155, 207, 313, 430, 608, 867, 1239, 1670, 2313, 3220, 4483

Offset

1,2

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.

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

Links

Table of n, a(n) for n=1..25.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

The a(1) = 1 through a(8) = 15 necklace compositions:
  (1)  (2)   (3)   (4)    (5)    (6)    (7)     (8)
       (11)  (12)  (13)   (14)   (15)   (16)    (17)
                   (22)   (23)   (24)   (25)    (26)
                   (112)  (113)  (33)   (34)    (35)
                          (122)  (114)  (115)   (44)
                                 (132)  (124)   (116)
                                        (133)   (125)
                                        (142)   (134)
                                        (223)   (143)
                                        (1132)  (152)
                                                (224)
                                                (233)
                                                (1124)
                                                (1142)
                                                (1322)

Mathematica

neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@Abs[Differences[#]]&&neckQ[#]&]], {n, 15}]

Crossrefs

Cf. A000079, A008965, A235998, A242882, A325325, A325352, A325404, A325468, A325552, A325554, A325556.

Sequence in context: A056219 A085140 A232166 * A138883 A257632 A325554

Adjacent sequences: A325552 A325553 A325554 * A325556 A325557 A325558

Keyword

nonn,more

Author

Gus Wiseman, May 11 2019

Status

approved