A325549 Number of necklace compositions of n with distinct circular differences.

1, 1, 2, 3, 5, 4, 10, 16, 23, 34, 53, 66, 113, 164, 262, 380, 567, 821, 1217, 1778, 2702, 3919, 5760, 8520, 12375

Offset

1,3

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 circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

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) = 16 necklace compositions:
  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)
            (12)  (13)   (14)   (15)   (16)    (17)
                  (112)  (23)   (24)   (25)    (26)
                         (113)  (114)  (34)    (35)
                         (122)         (115)   (116)
                                       (124)   (125)
                                       (133)   (134)
                                       (142)   (143)
                                       (223)   (152)
                                       (1213)  (224)
                                               (233)
                                               (1124)
                                               (1142)
                                               (1214)
                                               (11213)
                                               (11312)

Mathematica

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

Crossrefs

Cf. A000079, A000740, A008965, A034297, A059966, A070211, A318728, A318748, A320348, A325545, A325551, A325554, A325556.

Sequence in context: A254669 A227913 A079521 * A370895 A112060 A084933

Adjacent sequences: A325546 A325547 A325548 * A325550 A325551 A325552

Keyword

nonn,more

Author

Gus Wiseman, May 10 2019

Status

approved