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

Author

Gus Wiseman, May 11 2019

Keywords

• nonn
• more

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).

Examples

			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)
		

Links

• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

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

Programs

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}]