A325556 Number of necklace compositions of n with distinct circular differences up to sign.

1, 1, 1, 1, 1, 1, 3, 7, 9, 13, 25, 27, 51, 63, 95, 123, 179, 205, 305, 409, 559, 715, 1009, 1337, 1869

Offset

1,7

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(10) = 13 necklace compositions:
  (1)  (2)  (3)  (4)  (5)  (6)  (7)    (8)     (9)     (A)
                                (124)  (125)   (126)   (127)
                                (142)  (134)   (162)   (136)
                                       (143)   (1125)  (145)
                                       (152)   (1134)  (154)
                                       (1124)  (1143)  (163)
                                       (1142)  (1152)  (172)
                                               (1224)  (235)
                                               (1422)  (253)
                                                       (1126)
                                                       (1162)
                                                       (1225)
                                                       (1522)

Mathematica

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

Crossrefs

Cf. A000079, A000740, A008965, A235998, A318728, A325324, A325325, A325349, A325549, A325553, A325555, A325588, A325590.

Sequence in context: A109370 A018663 A110575 * A174840 A032664 A166733

Adjacent sequences: A325553 A325554 A325555 * A325557 A325558 A325559

Keyword

nonn,more

Author

Gus Wiseman, May 11 2019

Status

approved