OFFSET
1,2
COMMENTS
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations. Reversed necklaces are different from co-necklaces (A333764).
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
EXAMPLE
The sequence together with the corresponding reversed necklaces begins:
1: (1) 32: (6) 69: (4,2,1)
2: (2) 33: (5,1) 71: (4,1,1,1)
3: (1,1) 34: (4,2) 73: (3,3,1)
4: (3) 35: (4,1,1) 74: (3,2,2)
5: (2,1) 36: (3,3) 75: (3,2,1,1)
7: (1,1,1) 37: (3,2,1) 77: (3,1,2,1)
8: (4) 39: (3,1,1,1) 79: (3,1,1,1,1)
9: (3,1) 41: (2,3,1) 81: (2,4,1)
10: (2,2) 42: (2,2,2) 83: (2,3,1,1)
11: (2,1,1) 43: (2,2,1,1) 85: (2,2,2,1)
15: (1,1,1,1) 45: (2,1,2,1) 87: (2,2,1,1,1)
16: (5) 47: (2,1,1,1,1) 91: (2,1,2,1,1)
17: (4,1) 63: (1,1,1,1,1,1) 95: (2,1,1,1,1,1)
18: (3,2) 64: (7) 127: (1,1,1,1,1,1,1)
19: (3,1,1) 65: (6,1) 128: (8)
21: (2,2,1) 66: (5,2) 129: (7,1)
23: (2,1,1,1) 67: (5,1,1) 130: (6,2)
31: (1,1,1,1,1) 68: (4,3) 131: (6,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #1]}]&, Length[q]-1, 1, And];
Select[Range[100], neckQ[Reverse[stc[#]]]&]
CROSSREFS
The non-reversed version is A065609.
The dual version is A328595.
Binary necklaces are A000031.
Necklace compositions are A008965.
Necklaces covering an initial interval are A019536.
Numbers whose prime signature is a necklace are A329138.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of Lyndon factorization of reversed binary expansion is A329313.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Length of Lyndon factorization is A329312.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Length of co-Lyndon factorization is A334029.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 14 2020
STATUS
approved