%I #4 Apr 15 2020 09:52:57

%S 1,2,3,4,5,7,8,9,10,11,15,16,17,18,19,21,23,31,32,33,34,35,36,37,39,

%T 41,42,43,45,47,63,64,65,66,67,68,69,71,73,74,75,77,79,81,83,85,87,91,

%U 95,127,128,129,130,131,132,133,135,136,137,138,139,141,143

%N Numbers k such that the k-th composition in standard order is a reversed necklace.

%C A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations. Reversed necklaces are different from co-necklaces (A333764).

%C 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.

%e The sequence together with the corresponding reversed necklaces begins:

%e 1: (1) 32: (6) 69: (4,2,1)

%e 2: (2) 33: (5,1) 71: (4,1,1,1)

%e 3: (1,1) 34: (4,2) 73: (3,3,1)

%e 4: (3) 35: (4,1,1) 74: (3,2,2)

%e 5: (2,1) 36: (3,3) 75: (3,2,1,1)

%e 7: (1,1,1) 37: (3,2,1) 77: (3,1,2,1)

%e 8: (4) 39: (3,1,1,1) 79: (3,1,1,1,1)

%e 9: (3,1) 41: (2,3,1) 81: (2,4,1)

%e 10: (2,2) 42: (2,2,2) 83: (2,3,1,1)

%e 11: (2,1,1) 43: (2,2,1,1) 85: (2,2,2,1)

%e 15: (1,1,1,1) 45: (2,1,2,1) 87: (2,2,1,1,1)

%e 16: (5) 47: (2,1,1,1,1) 91: (2,1,2,1,1)

%e 17: (4,1) 63: (1,1,1,1,1,1) 95: (2,1,1,1,1,1)

%e 18: (3,2) 64: (7) 127: (1,1,1,1,1,1,1)

%e 19: (3,1,1) 65: (6,1) 128: (8)

%e 21: (2,2,1) 66: (5,2) 129: (7,1)

%e 23: (2,1,1,1) 67: (5,1,1) 130: (6,2)

%e 31: (1,1,1,1,1) 68: (4,3) 131: (6,1,1)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#1]}]&,Length[q]-1,1,And];

%t Select[Range[100],neckQ[Reverse[stc[#]]]&]

%Y The non-reversed version is A065609.

%Y The dual version is A328595.

%Y Binary necklaces are A000031.

%Y Necklace compositions are A008965.

%Y Necklaces covering an initial interval are A019536.

%Y Numbers whose prime signature is a necklace are A329138.

%Y Length of co-Lyndon factorization of binary expansion is A329312.

%Y Length of Lyndon factorization of reversed binary expansion is A329313.

%Y All of the following pertain to compositions in standard order (A066099):

%Y - Length is A000120.

%Y - Sum is A070939.

%Y - Runs are counted by A124767.

%Y - Rotational symmetries are counted by A138904.

%Y - Strict compositions are A233564.

%Y - Constant compositions are A272919.

%Y - Lyndon compositions are A275692.

%Y - Co-Lyndon compositions are A326774.

%Y - Aperiodic compositions are A328594.

%Y - Length of Lyndon factorization is A329312.

%Y - Rotational period is A333632.

%Y - Co-necklaces are A333764.

%Y - Length of co-Lyndon factorization is A334029.

%Y Cf. A000740, A001037, A027375, A059966, A211100, A302291, A328596, A329142, A333765, A333939, A333941.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 14 2020