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A325549 Number of necklace compositions of n with distinct circular differences. 7
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 (list; graph; refs; listen; history; text; internal format)
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 * A112060 A084933 A213900

Adjacent sequences:  A325546 A325547 A325548 * A325550 A325551 A325552

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, May 10 2019

STATUS

approved

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Last modified May 12 10:54 EDT 2021. Contains 343821 sequences. (Running on oeis4.)