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Numbers k such that the k-th composition in standard order is both a reversed necklace and a co-necklace.
5

%I #6 Apr 28 2020 00:14:43

%S 0,1,2,3,4,5,7,8,9,10,11,15,16,17,18,19,21,23,31,32,33,34,35,36,37,39,

%T 42,43,45,47,63,64,65,66,67,68,69,71,73,74,75,77,79,85,87,91,95,127,

%U 128,129,130,131,132,133,135,136,137,138,139,141,143,146,147

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

%C A necklace is a finite sequence of positive integers that is lexicographically less than or equal to any cyclic rotation. Co-necklaces are defined similarly, except with greater instead of less.

%C The k-th composition in standard order (graded reverse-lexicographic, 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 of all reversed necklace co-necklaces begins:

%e 0: () 31: (1,1,1,1,1) 69: (4,2,1)

%e 1: (1) 32: (6) 71: (4,1,1,1)

%e 2: (2) 33: (5,1) 73: (3,3,1)

%e 3: (1,1) 34: (4,2) 74: (3,2,2)

%e 4: (3) 35: (4,1,1) 75: (3,2,1,1)

%e 5: (2,1) 36: (3,3) 77: (3,1,2,1)

%e 7: (1,1,1) 37: (3,2,1) 79: (3,1,1,1,1)

%e 8: (4) 39: (3,1,1,1) 85: (2,2,2,1)

%e 9: (3,1) 42: (2,2,2) 87: (2,2,1,1,1)

%e 10: (2,2) 43: (2,2,1,1) 91: (2,1,2,1,1)

%e 11: (2,1,1) 45: (2,1,2,1) 95: (2,1,1,1,1,1)

%e 15: (1,1,1,1) 47: (2,1,1,1,1) 127: (1,1,1,1,1,1,1)

%e 16: (5) 63: (1,1,1,1,1,1) 128: (8)

%e 17: (4,1) 64: (7) 129: (7,1)

%e 18: (3,2) 65: (6,1) 130: (6,2)

%e 19: (3,1,1) 66: (5,2) 131: (6,1,1)

%e 21: (2,2,1) 67: (5,1,1) 132: (5,3)

%e 23: (2,1,1,1) 68: (4,3) 133: (5,2,1)

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

%t neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];

%t coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];

%t Select[Range[0,100],neckQ[Reverse[stc[#]]]&&coneckQ[stc[#]]&]

%Y The aperiodic case is A334266.

%Y Compositions of this type are counted by A334271.

%Y Normal sequences of this type are counted by A334272.

%Y Another ranking of the same compositions is A334274 (binary expansion).

%Y Binary (or reversed binary) necklaces are counted by A000031.

%Y Necklace compositions are counted by A008965.

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

%Y - Necklaces are A065609.

%Y - Reversed necklaces are A333943.

%Y - Co-necklaces are A333764.

%Y - Reversed co-necklaces are A328595.

%Y - Lyndon words are A275692.

%Y - Co-Lyndon words are A326774.

%Y - Reversed Lyndon words are A334265.

%Y - Reversed co-Lyndon words are A328596.

%Y - Aperiodic compositions are A328594.

%Y Cf. A019536, A034691, A059966, A060223, A329138, A334269, A334270.

%K nonn

%O 1,3

%A _Gus Wiseman_, Apr 25 2020