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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
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).
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}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, May 10 2019
STATUS
approved