A325554 Number of necklace compositions of n with distinct differences.

1, 2, 2, 4, 5, 6, 11, 18, 26, 38, 60, 90, 139, 213, 329, 501, 747, 1144, 1712, 2548, 3836, 5732, 8442, 12654, 18624

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) = 18 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)
                                        (1213)  (224)
                                                (233)
                                                (1124)
                                                (1142)
                                                (1214)
                                                (1322)
                                                (11213)
                                                (11312)

Mathematica

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

Crossrefs

Cf. A000079, A008965, A235998, A242882, A318728, A325325, A325404, A325468, A325545, A325549, A325550, A325555.

Sequence in context: A325555 A138883 A257632 * A367962 A107849 A053036

Adjacent sequences: A325551 A325552 A325553 * A325555 A325556 A325557

Keyword

nonn,more

Author

Gus Wiseman, May 11 2019

Status

approved