cp's OEIS Frontend

A334267 Numbers k such that the k-th composition in standard order is both a Lyndon word and a reversed co-Lyndon word.

%I A334267 #9 Apr 25 2020 08:40:45
%S A334267 0,1,2,4,6,8,12,14,16,20,24,26,28,30,32,40,48,52,56,58,60,62,64,72,80,
%T A334267 84,96,100,104,106,108,112,116,118,120,122,124,126,128,144,160,164,
%U A334267 168,192,200,208,212,216,218,224,228,232,234,236,240,244,246,248,250
%N A334267 Numbers k such that the k-th composition in standard order is both a Lyndon word and a reversed co-Lyndon word.
%C A334267 Also numbers whose binary expansion is both a reversed Lyndon word and a co-Lyndon word.
%C A334267 A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations. Co-Lyndon is defined similarly, except with strictly greater instead of strictly less.
%C A334267 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.
%F A334267 Intersection of A275692 and A328596.
%e A334267 The sequence of all reversed co-Lyndon Lyndon words begins:
%e A334267     0: ()            56: (1,1,4)        124: (1,1,1,1,3)
%e A334267     1: (1)           58: (1,1,2,2)      126: (1,1,1,1,1,2)
%e A334267     2: (2)           60: (1,1,1,3)      128: (8)
%e A334267     4: (3)           62: (1,1,1,1,2)    144: (3,5)
%e A334267     6: (1,2)         64: (7)            160: (2,6)
%e A334267     8: (4)           72: (3,4)          164: (2,3,3)
%e A334267    12: (1,3)         80: (2,5)          168: (2,2,4)
%e A334267    14: (1,1,2)       84: (2,2,3)        192: (1,7)
%e A334267    16: (5)           96: (1,6)          200: (1,3,4)
%e A334267    20: (2,3)        100: (1,3,3)        208: (1,2,5)
%e A334267    24: (1,4)        104: (1,2,4)        212: (1,2,2,3)
%e A334267    26: (1,2,2)      106: (1,2,2,2)      216: (1,2,1,4)
%e A334267    28: (1,1,3)      108: (1,2,1,3)      218: (1,2,1,2,2)
%e A334267    30: (1,1,1,2)    112: (1,1,5)        224: (1,1,6)
%e A334267    32: (6)          116: (1,1,2,3)      228: (1,1,3,3)
%e A334267    40: (2,4)        118: (1,1,2,1,2)    232: (1,1,2,4)
%e A334267    48: (1,5)        120: (1,1,1,4)      234: (1,1,2,2,2)
%e A334267    52: (1,2,3)      122: (1,1,1,2,2)    236: (1,1,2,1,3)
%e A334267 The sequence of terms together with their binary expansions and binary indices begins:
%e A334267     0:      0 ~ {}            56:  111000 ~ {4,5,6}
%e A334267     1:      1 ~ {1}           58:  111010 ~ {2,4,5,6}
%e A334267     2:     10 ~ {2}           60:  111100 ~ {3,4,5,6}
%e A334267     4:    100 ~ {3}           62:  111110 ~ {2,3,4,5,6}
%e A334267     6:    110 ~ {2,3}         64: 1000000 ~ {7}
%e A334267     8:   1000 ~ {4}           72: 1001000 ~ {4,7}
%e A334267    12:   1100 ~ {3,4}         80: 1010000 ~ {5,7}
%e A334267    14:   1110 ~ {2,3,4}       84: 1010100 ~ {3,5,7}
%e A334267    16:  10000 ~ {5}           96: 1100000 ~ {6,7}
%e A334267    20:  10100 ~ {3,5}        100: 1100100 ~ {3,6,7}
%e A334267    24:  11000 ~ {4,5}        104: 1101000 ~ {4,6,7}
%e A334267    26:  11010 ~ {2,4,5}      106: 1101010 ~ {2,4,6,7}
%e A334267    28:  11100 ~ {3,4,5}      108: 1101100 ~ {3,4,6,7}
%e A334267    30:  11110 ~ {2,3,4,5}    112: 1110000 ~ {5,6,7}
%e A334267    32: 100000 ~ {6}          116: 1110100 ~ {3,5,6,7}
%e A334267    40: 101000 ~ {4,6}        118: 1110110 ~ {2,3,5,6,7}
%e A334267    48: 110000 ~ {5,6}        120: 1111000 ~ {4,5,6,7}
%e A334267    52: 110100 ~ {3,5,6}      122: 1111010 ~ {2,4,5,6,7}
%t A334267 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A334267 lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
%t A334267 colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
%t A334267 Select[Range[0,100],colynQ[Reverse[stc[#]]]&&lynQ[stc[#]]&]
%Y A334267 Compositions of this type are counted by A334269.
%Y A334267 Normal sequences of this type are counted by A334270.
%Y A334267 Necklaces of this type are counted by A334271.
%Y A334267 Necklaces of this type are ranked by A334274.
%Y A334267 Binary (or reversed binary) Lyndon words are counted by A001037.
%Y A334267 Lyndon compositions are counted by A059966.
%Y A334267 Lyndon words whose reverse is not co-Lyndon are counted by A329324
%Y A334267 Reversed Lyndon co-Lyndon compositions are ranked by A334266.
%Y A334267 All of the following pertain to compositions in standard order (A066099):
%Y A334267 - Length is A000120.
%Y A334267 - Necklaces are A065609.
%Y A334267 - Sum is A070939.
%Y A334267 - Reverse is A228351 (triangle).
%Y A334267 - Strict compositions are A233564.
%Y A334267 - Constant compositions are A272919.
%Y A334267 - Lyndon words are A275692.
%Y A334267 - Reversed Lyndon words are A334265.
%Y A334267 - Co-Lyndon words are A326774.
%Y A334267 - Reversed co-Lyndon words are A328596.
%Y A334267 - Length of Lyndon factorization is A329312.
%Y A334267 - Length of Lyndon factorization of reverse is A334297.
%Y A334267 - Length of co-Lyndon factorization is A334029.
%Y A334267 - Length of co-Lyndon factorization of reverse is A329313.
%Y A334267 - Distinct rotations are counted by A333632.
%Y A334267 - Lyndon factorizations are counted by A333940.
%Y A334267 - Co-Lyndon factorizations are counted by A333765.
%Y A334267 Cf. A000740, A008965, A328594, A328595, A333764, A333943, A334272, A334273.
%K A334267 nonn
%O A334267 1,3
%A A334267 _Gus Wiseman_, Apr 22 2020