A329313 -id:A329313 - cp's OEIS frontend

A334266 Numbers k such that the k-th composition in standard order is both a reversed Lyndon word and a co-Lyndon word.

0, 1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 21, 23, 32, 33, 34, 35, 37, 39, 43, 47, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 135, 137, 138, 139, 141, 143, 146, 147, 149, 151, 155, 159, 171, 173, 175, 183, 191

Offset: 1

Gus Wiseman, Apr 22 2020

nonn

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.

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.

			The sequence of all reversed Lyndon co-Lyndon words begins:
    0: ()            37: (3,2,1)         91: (2,1,2,1,1)
    1: (1)           39: (3,1,1,1)       95: (2,1,1,1,1,1)
    2: (2)           43: (2,2,1,1)      128: (8)
    4: (3)           47: (2,1,1,1,1)    129: (7,1)
    5: (2,1)         64: (7)            130: (6,2)
    8: (4)           65: (6,1)          131: (6,1,1)
    9: (3,1)         66: (5,2)          132: (5,3)
   11: (2,1,1)       67: (5,1,1)        133: (5,2,1)
   16: (5)           68: (4,3)          135: (5,1,1,1)
   17: (4,1)         69: (4,2,1)        137: (4,3,1)
   18: (3,2)         71: (4,1,1,1)      138: (4,2,2)
   19: (3,1,1)       73: (3,3,1)        139: (4,2,1,1)
   21: (2,2,1)       74: (3,2,2)        141: (4,1,2,1)
   23: (2,1,1,1)     75: (3,2,1,1)      143: (4,1,1,1,1)
   32: (6)           77: (3,1,2,1)      146: (3,3,2)
   33: (5,1)         79: (3,1,1,1,1)    147: (3,3,1,1)
   34: (4,2)         85: (2,2,2,1)      149: (3,2,2,1)
   35: (4,1,1)       87: (2,2,1,1,1)    151: (3,2,1,1,1)

The version for binary expansion is A334267.
Compositions of this type are counted by A334269.
Normal sequences of this type are counted by A334270.
Necklace compositions of this type are counted by A334271.
Binary Lyndon words are counted by A001037.
Lyndon compositions are counted by A059966.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Reversed Lyndon words are A334265.
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Length of Lyndon factorization of reverse is A334297.
- Length of co-Lyndon factorization is A334029.
- Length of co-Lyndon factorization of reverse is A329313.
- Distinct rotations are counted by A333632.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
Cf. A000740, A008965, A065609, A329324, A333764, A333943, A334272, A334297.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Select[Range[0,100],lynQ[Reverse[stc[#]]]&&colynQ[stc[#]]&]

Intersection of A334265 and A326774.

A334269 Number of compositions of n that are both a reversed Lyndon word and a co-Lyndon word.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 16, 23, 40, 62, 110, 169, 302, 492, 856, 1454, 2572, 4428, 7914, 13935, 25036, 44842, 81298, 147149, 268952, 491746, 904594, 1667091, 3085950, 5723367, 10652544, 19865887, 37150314, 69608939, 130723184, 245935633, 463590444, 875306913, 1655451592, 3135613649, 5948011978, 11298215516

Offset: 1

Gus Wiseman, Apr 24 2020

nonn

Also the number of compositions of n that are both a Lyndon word and a reversed co-Lyndon word.
A composition of n is a finite sequence of positive integers summing to n.
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.

			The a(1) = 1 through a(7) = 16 compositions:
  (1)  (2)  (3)   (4)    (5)     (6)      (7)
            (21)  (31)   (32)    (42)     (43)
                  (211)  (41)    (51)     (52)
                         (221)   (321)    (61)
                         (311)   (411)    (322)
                         (2111)  (2211)   (331)
                                 (3111)   (421)
                                 (21111)  (511)
                                          (2221)
                                          (3121)
                                          (3211)
                                          (4111)
                                          (21211)
                                          (22111)
                                          (31111)
                                          (211111)

The version for binary expansion is A334267.
Compositions of this type are ranked by A334266.
Normal sequences of this type are counted by A334270.
Necklace compositions of this type are counted by A334271.
Aperiodic compositions are counted by A000740.
Binary Lyndon words are counted by A001037.
Necklace compositions are counted by A008965.
Normal Lyndon words are counted by A060223.
Lyndon compositions are counted by A059966.
All of the following pertain to compositions in standard order (A066099):
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed Lyndon words are A334265.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Length of co-Lyndon factorization is A334029.
- Length of Lyndon factorization of reverse is A334297.
- Length of co-Lyndon factorization of reverse is A329313.
- Lyndon factorizations are counted by A333940.
- Co-Lyndon factorizations are counted by A333765.
- Aperiodic compositions are A328594.
- Distinct rotations are counted by A333632.
Cf. A034691, A065609, A275692, A328596, A329141, A329324, A329326, A334266, A334272, A334273, A334274.

lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],lynQ[Reverse[#]]&&colynQ[#]&]],{n,0,15}]

Offset corrected and a(21)-a(42) from Bert Dobbelaere, Apr 26 2020

A329327 Numbers whose binary expansion has Lyndon factorization of length 2 (the minimum for n > 1).

Original entry on oeis.org

2, 3, 5, 9, 11, 17, 19, 23, 33, 35, 37, 39, 43, 47, 65, 67, 69, 71, 75, 77, 79, 87, 95, 129, 131, 133, 135, 137, 139, 141, 143, 147, 149, 151, 155, 157, 159, 171, 175, 183, 191, 257, 259, 261, 263, 265, 267, 269, 271, 275, 277, 279, 281, 283, 285, 287, 293

Offset: 1

Gus Wiseman, Nov 12 2019

nonn

First differs from A329357 in having 77 and lacking 83.

Also numbers whose decapitated binary expansion is a Lyndon word (see also A329401).

			The binary expansion of each term together with its Lyndon factorization begins:
   2:      (10) = (1)(0)
   3:      (11) = (1)(1)
   5:     (101) = (1)(01)
   9:    (1001) = (1)(001)
  11:    (1011) = (1)(011)
  17:   (10001) = (1)(0001)
  19:   (10011) = (1)(0011)
  23:   (10111) = (1)(0111)
  33:  (100001) = (1)(00001)
  35:  (100011) = (1)(00011)
  37:  (100101) = (1)(00101)
  39:  (100111) = (1)(00111)
  43:  (101011) = (1)(01011)
  47:  (101111) = (1)(01111)
  65: (1000001) = (1)(000001)
  67: (1000011) = (1)(000011)
  69: (1000101) = (1)(000101)
  71: (1000111) = (1)(000111)
  75: (1001011) = (1)(001011)
  77: (1001101) = (1)(001101)

Positions of 2's in A211100.
Positions of rows of length 2 in A329314.
The "co-" and reversed version is A329357.
Binary Lyndon words are counted by A001037 and ranked by A102659.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A059966, A060223, A211097, A275692, A328594, A328595, A328596, A329131, A329313, A329325, A329326, A339608.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#1]]&]]]];
Select[Range[100],Length[lynfac[IntegerDigits[#,2]]]==2&]

a(n) = A339608(n) + 1. - Harald Korneliussen, Mar 12 2020

A333941 Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 2, 3, 0, 0, 2, 4, 6, 4, 0, 0, 4, 6, 9, 8, 5, 0, 0, 2, 6, 15, 20, 15, 6, 0, 0, 4, 8, 24, 32, 35, 18, 7, 0, 0, 3, 10, 27, 56, 70, 54, 28, 8, 0, 0, 4, 12, 42, 84, 125, 120, 84, 32, 9, 0, 0, 2, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset: 0

Gus Wiseman, Apr 16 2020

A composition of n is a finite sequence of positive integers summing to n.

			Triangle begins:
   1
   0   1
   0   2   0
   0   2   2   0
   0   3   2   3   0
   0   2   4   6   4   0
   0   4   6   9   8   5   0
   0   2   6  15  20  15   6   0
   0   4   8  24  32  35  18   7   0
   0   3  10  27  56  70  54  28   8   0
   0   4  12  42  84 125 120  84  32   9   0
   0   2  10  45 120 210 252 210 120  45  10   0
   0   6  18  66 168 335 450 462 320 162  50  11   0
Row n = 6 counts the following compositions (empty columns indicated by dots):
  .  (6)       (15)    (114)  (1113)  (11112)  .
     (33)      (24)    (123)  (1122)  (11121)
     (222)     (42)    (132)  (1131)  (11211)
     (111111)  (51)    (141)  (1221)  (12111)
               (1212)  (213)  (1311)  (21111)
               (2121)  (231)  (2112)
                       (312)  (2211)
                       (321)  (3111)
                       (411)

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Column k = 1 is A000005.
Row sums are A011782.
Diagonal T(2n,n) is A045630(n).
The strict version is A072574.
A version counting runs is A238279.
Column k = n - 1 is A254667.
Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
Period of binary expansion is A302291.
Numbers whose prime signature is aperiodic are A329139.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Rotational symmetries are counted by A138904.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Reversed necklaces are A333943.
Cf. A000031, A001037, A008965, A019536, A211100, A291166, A328595, A328596, A329312, A329313, A329326.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Function[c,Length[Union[Array[RotateRight[c,#]&,Length[c]]]]==k]]],{n,0,10},{k,0,n}]

T(n,k)=if(n==0, k==0, sumdiv(n, m, sumdiv(gcd(k,m), d, moebius(d)*binomial(m/d-1, k/d-1)))) \\ Andrew Howroyd, Jan 19 2023

T(n,k) = Sum_{m|n} Sum_{d|gcd(k,m)} mu(d)*binomial(m/d-1, k/d-1) for n > 0. - Andrew Howroyd, Jan 19 2023

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

Original entry on oeis.org

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, 84, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 128, 144, 160, 164, 168, 192, 200, 208, 212, 216, 218, 224, 228, 232, 234, 236, 240, 244, 246, 248, 250

Offset: 1

Gus Wiseman, Apr 22 2020

nonn

Also numbers whose binary expansion is both a reversed Lyndon word and a co-Lyndon word.
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.
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.

			The sequence of all reversed co-Lyndon Lyndon words begins:
    0: ()            56: (1,1,4)        124: (1,1,1,1,3)
    1: (1)           58: (1,1,2,2)      126: (1,1,1,1,1,2)
    2: (2)           60: (1,1,1,3)      128: (8)
    4: (3)           62: (1,1,1,1,2)    144: (3,5)
    6: (1,2)         64: (7)            160: (2,6)
    8: (4)           72: (3,4)          164: (2,3,3)
   12: (1,3)         80: (2,5)          168: (2,2,4)
   14: (1,1,2)       84: (2,2,3)        192: (1,7)
   16: (5)           96: (1,6)          200: (1,3,4)
   20: (2,3)        100: (1,3,3)        208: (1,2,5)
   24: (1,4)        104: (1,2,4)        212: (1,2,2,3)
   26: (1,2,2)      106: (1,2,2,2)      216: (1,2,1,4)
   28: (1,1,3)      108: (1,2,1,3)      218: (1,2,1,2,2)
   30: (1,1,1,2)    112: (1,1,5)        224: (1,1,6)
   32: (6)          116: (1,1,2,3)      228: (1,1,3,3)
   40: (2,4)        118: (1,1,2,1,2)    232: (1,1,2,4)
   48: (1,5)        120: (1,1,1,4)      234: (1,1,2,2,2)
   52: (1,2,3)      122: (1,1,1,2,2)    236: (1,1,2,1,3)
The sequence of terms together with their binary expansions and binary indices begins:
    0:      0 ~ {}            56:  111000 ~ {4,5,6}
    1:      1 ~ {1}           58:  111010 ~ {2,4,5,6}
    2:     10 ~ {2}           60:  111100 ~ {3,4,5,6}
    4:    100 ~ {3}           62:  111110 ~ {2,3,4,5,6}
    6:    110 ~ {2,3}         64: 1000000 ~ {7}
    8:   1000 ~ {4}           72: 1001000 ~ {4,7}
   12:   1100 ~ {3,4}         80: 1010000 ~ {5,7}
   14:   1110 ~ {2,3,4}       84: 1010100 ~ {3,5,7}
   16:  10000 ~ {5}           96: 1100000 ~ {6,7}
   20:  10100 ~ {3,5}        100: 1100100 ~ {3,6,7}
   24:  11000 ~ {4,5}        104: 1101000 ~ {4,6,7}
   26:  11010 ~ {2,4,5}      106: 1101010 ~ {2,4,6,7}
   28:  11100 ~ {3,4,5}      108: 1101100 ~ {3,4,6,7}
   30:  11110 ~ {2,3,4,5}    112: 1110000 ~ {5,6,7}
   32: 100000 ~ {6}          116: 1110100 ~ {3,5,6,7}
   40: 101000 ~ {4,6}        118: 1110110 ~ {2,3,5,6,7}
   48: 110000 ~ {5,6}        120: 1111000 ~ {4,5,6,7}
   52: 110100 ~ {3,5,6}      122: 1111010 ~ {2,4,5,6,7}

Compositions of this type are counted by A334269.
Normal sequences of this type are counted by A334270.
Necklaces of this type are counted by A334271.
Necklaces of this type are ranked by A334274.
Binary (or reversed binary) Lyndon words are counted by A001037.
Lyndon compositions are counted by A059966.
Lyndon words whose reverse is not co-Lyndon are counted by A329324
Reversed Lyndon co-Lyndon compositions are ranked by A334266.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Reversed Lyndon words are A334265.
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Length of Lyndon factorization of reverse is A334297.
- Length of co-Lyndon factorization is A334029.
- Length of co-Lyndon factorization of reverse is A329313.
- Distinct rotations are counted by A333632.
- Lyndon factorizations are counted by A333940.
- Co-Lyndon factorizations are counted by A333765.
Cf. A000740, A008965, A328594, A328595, A333764, A333943, A334272, A334273.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Select[Range[0,100],colynQ[Reverse[stc[#]]]&&lynQ[stc[#]]&]

Intersection of A275692 and A328596.

A334270 Number of sequences of length n that cover an initial interval of positive integers and are both a reversed Lyndon word and a co-Lyndon word.

Original entry on oeis.org

1, 1, 1, 3, 10, 42, 224, 1505, 12380, 120439

Offset: 0

Gus Wiseman, Apr 24 2020

Also the number of sequences of length n that cover an initial interval of positive integers and are both a Lyndon word and a reversed co-Lyndon word.

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.

			The a(1) = 1 through a(4) = 10 normal sequences:
  (1)  (2,1)  (2,1,1)  (2,1,1,1)
              (2,2,1)  (2,2,1,1)
              (3,2,1)  (2,2,2,1)
                       (3,1,2,1)
                       (3,2,1,1)
                       (3,2,2,1)
                       (3,2,3,1)
                       (3,3,2,1)
                       (4,2,3,1)
                       (4,3,2,1)

These compositions are ranked by A334266 (standard) and A334267 (binary).
Compositions of this type are counted by A334269.
Necklace compositions of this type are counted by A334271.
Dominated by A334272 (the necklace version).
Normal sequences are counted by A000670.
Binary (or reversed binary) Lyndon words are counted by A001037.
Lyndon compositions are counted by A059966.
Normal Lyndon words are counted by A060223.
Normal sequences by length and Lyndon factorization length are A296372.
All of the following pertain to compositions in standard order (A066099):
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed Lyndon words are A334265.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Length of co-Lyndon factorization is A334029.
- Length of Lyndon factorization of reverse is A334297.
- Length of co-Lyndon factorization of reverse is A329313.
Cf. A008965, A034691, A329324, A329398, A334273.

lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],lynQ[Reverse[#]]&&colynQ[#]&]],{n,0,6}]

A334297 Length of the Lyndon factorization of the reversed n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 6, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2

Offset: 0

Gus Wiseman, Apr 25 2020

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

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.

			The 12345th composition is (1,7,1,1,3,1), with reverse (1,3,1,1,7,1), with Lyndon factorization ((1),(1,3),(1,1,7)), so a(12345) = 3.

The non-reversed version is A329312.
The version for binary indices is A329313 (also the "co-" version).
Positions of 1's are A334265 (reversed Lyndon words).
Binary Lyndon words are counted by A001037 and ranked by A102659.
Lyndon compositions are counted by A059966 and ranked by A275692.
Normal Lyndon sequences are counted by A060223 (row sums of A296372).
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Aperiodic compositions are A328594.
- Distinct rotations are counted by A333632.
- Lyndon factorizations are counted by A333940.
- Length of co-Lyndon factorization is A334029.
Cf. A027375, A211097, A211100, A328595, A329314, A329317.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
Table[Length[lynfac[Reverse[stc[n]]]],{n,0,100}]

A329396 Numbers k such that the co-Lyndon factorization of the binary expansion of k is uniform.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 38, 40, 42, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 72, 80, 84, 96, 98, 100, 104, 106, 108, 112, 114, 116, 118, 120, 122, 124, 126, 127, 128, 136, 140, 142, 144, 160, 164, 168, 170, 192

Offset: 1

Gus Wiseman, Nov 13 2019

nonn

The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

A sequence of words is uniform if they all have the same length.

			The sequence of terms together with their co-Lyndon factorizations begins:
   1:      (1) = (1)
   2:     (10) = (10)
   3:     (11) = (1)(1)
   4:    (100) = (100)
   6:    (110) = (110)
   7:    (111) = (1)(1)(1)
   8:   (1000) = (1000)
  10:   (1010) = (10)(10)
  12:   (1100) = (1100)
  14:   (1110) = (1110)
  15:   (1111) = (1)(1)(1)(1)
  16:  (10000) = (10000)
  20:  (10100) = (10100)
  24:  (11000) = (11000)
  26:  (11010) = (11010)
  28:  (11100) = (11100)
  30:  (11110) = (11110)
  31:  (11111) = (1)(1)(1)(1)(1)
  32: (100000) = (100000)
  36: (100100) = (100)(100)
  38: (100110) = (100)(110)
  40: (101000) = (101000)
  42: (101010) = (10)(10)(10)

Numbers whose binary expansion has uniform Lyndon factorization are A023758.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Numbers whose trimmed binary expansion has Lyndon and co-Lyndon factorizations of equal lengths are A329395.
Cf. A001037, A059966, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329398.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Select[Range[100],SameQ@@Length/@colynfac[IntegerDigits[#,2]]&]

A329399 Numbers whose reversed binary expansion has uniform Lyndon factorization.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 38, 40, 42, 44, 48, 52, 54, 56, 58, 60, 62, 63, 64, 72, 80, 84, 88, 92, 96, 100, 104, 106, 108, 112, 116, 118, 120, 122, 124, 126, 127, 128, 136, 140, 142, 144, 152, 160, 164, 168, 170

Offset: 1

Gus Wiseman, Nov 13 2019

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

A sequence of words is uniform if they all have the same length.

			The sequence of terms together with their reversed binary expansions and Lyndon factorizations begins:
   1:      (1) = (1)
   2:     (01) = (01)
   3:     (11) = (1)(1)
   4:    (001) = (001)
   6:    (011) = (011)
   7:    (111) = (1)(1)(1)
   8:   (0001) = (0001)
  10:   (0101) = (01)(01)
  12:   (0011) = (0011)
  14:   (0111) = (0111)
  15:   (1111) = (1)(1)(1)(1)
  16:  (00001) = (00001)
  20:  (00101) = (00101)
  24:  (00011) = (00011)
  26:  (01011) = (01011)
  28:  (00111) = (00111)
  30:  (01111) = (01111)
  31:  (11111) = (1)(1)(1)(1)(1)
  32: (000001) = (000001)
  36: (001001) = (001)(001)
  38: (011001) = (011)(001)
  40: (000101) = (000101)
  42: (010101) = (01)(01)(01)
  44: (001101) = (001101)
  48: (000011) = (000011)

Numbers whose binary expansion has uniform Lyndon factorization and uniform co-Lyndon factorization are A023758.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Numbers whose trimmed binary expansion has Lyndon and co-Lyndon factorizations of equal lengths are A329395.
Cf. A001037, A059966, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329396.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
Select[Range[100],SameQ@@Length/@lynfac[Reverse[IntegerDigits[#,2]]]&]

A329400 Length of the co-Lyndon factorization of the binary expansion of n with the most significant (first) digit removed.

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 3, 3, 2, 3, 1, 2, 1, 3, 4, 4, 3, 4, 2, 3, 2, 4, 1, 2, 2, 3, 1, 2, 1, 4, 5, 5, 4, 5, 3, 4, 3, 5, 2, 3, 3, 4, 2, 3, 2, 5, 1, 2, 2, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 1, 5, 6, 6, 5, 6, 4, 5, 4, 6, 3, 4, 4, 5, 3, 4, 3, 6, 2, 3, 3, 4, 2, 4, 3, 5

Offset: 1

Gus Wiseman, Nov 16 2019

nonn

The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

			Decapitated binary expansions of 1..20 together with their co-Lyndon factorizations:
   1:     () =
   2:    (0) = (0)
   3:    (1) = (1)
   4:   (00) = (0)(0)
   5:   (01) = (0)(1)
   6:   (10) = (10)
   7:   (11) = (1)(1)
   8:  (000) = (0)(0)(0)
   9:  (001) = (0)(0)(1)
  10:  (010) = (0)(10)
  11:  (011) = (0)(1)(1)
  12:  (100) = (100)
  13:  (101) = (10)(1)
  14:  (110) = (110)
  15:  (111) = (1)(1)(1)
  16: (0000) = (0)(0)(0)(0)
  17: (0001) = (0)(0)(0)(1)
  18: (0010) = (0)(0)(10)
  19: (0011) = (0)(0)(1)(1)
  20: (0100) = (0)(100)

The non-"co" version is A211097.
The version involving all digits is A329312.
Lyndon and co-Lyndon compositions are (both) counted by A059966.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Numbers whose decapitated binary expansion is co-Lyndon are A329401.
Cf. A001037, A060223, A102659, A211100, A329313, A329318, A329326, A329395.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Table[If[n==0,0,Length[colynfac[Rest[IntegerDigits[n,2]]]]],{n,30}]

cp's OEIS Frontend

A334266 Numbers k such that the k-th composition in standard order is both a reversed Lyndon word and a co-Lyndon word.

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A334269 Number of compositions of n that are both a reversed Lyndon word and a co-Lyndon word.

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A329327 Numbers whose binary expansion has Lyndon factorization of length 2 (the minimum for n > 1).

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A333941 Triangle read by rows where T(n,k) is the number of compositions of n with rotational period k.

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

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A334270 Number of sequences of length n that cover an initial interval of positive integers and are both a reversed Lyndon word and a co-Lyndon word.

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A334297 Length of the Lyndon factorization of the reversed n-th composition in standard order.

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A329396 Numbers k such that the co-Lyndon factorization of the binary expansion of k is uniform.

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