A211097 -id:A211097 - cp's OEIS frontend

A211100 Number of factors in Lyndon factorization of binary expansion of n.

1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 4, 3, 4, 4, 5, 2, 3, 2, 4, 3, 3, 2, 5, 3, 4, 3, 5, 4, 5, 5, 6, 2, 3, 2, 4, 2, 3, 2, 5, 3, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 5, 4, 4, 3, 6, 4, 5, 4, 6, 5, 6, 6, 7, 2, 3, 2, 4, 2, 3, 2, 5, 3, 3, 2, 4, 2, 3, 2, 6, 3, 4, 3, 5, 4, 3, 2, 5, 3, 4, 3, 4, 3, 3, 2, 7, 3, 4, 3, 5, 3, 4, 3, 6, 4, 5, 3, 5, 4, 4, 3, 7, 4, 5, 4, 6, 5, 5, 4, 7

Offset: 0

N. J. A. Sloane, Mar 31 2012

nonn

Any binary word has a unique factorization as a product of nonincreasing Lyndon words (see Lothaire). a(n) = number of factors in Lyndon factorization of binary expansion of n.
It appears that a(n) = k for the first time when n = 2^(k-1)+1.
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. - Gus Wiseman, Nov 12 2019

			n=25 has binary expansion 11001, which has Lyndon factorization (1)(1)(001) with three factors, so a(25) = 3.
Here are the Lyndon factorizations for small values of n:
.0.
.1.
.1.0.
.1.1.
.1.0.0.
.1.01.
.1.1.0.
.1.1.1.
.1.0.0.0.
.1.001.
.1.01.0.
.1.011.
.1.1.0.0.
...

M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. See Theorem 5.1.5, p. 67.
G. Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Maple programs for A211100 etc.

Cf. A001037 (number of Lyndon words of length m); A102659 (list thereof).
A211095 and A211096 give information about the smallest (or rightmost) factor. Cf. A211097, A211098, A211099.
Row-lengths of A329314.
The "co-" version is A329312.
Positions of 2's are A329327.
The reversed version is A329313.
The inverted version is A329312.
Ignoring the first digit gives A211097.
Cf. A059966, A060223, A275692, A296658, A328594, A328595, A328596, A329131, A329325, A329326.

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,#]]&]]]];
Table[Length[lynfac[IntegerDigits[n,2]]],{n,0,30}] (* Gus Wiseman, Nov 12 2019 *)

A329312 Length of the co-Lyndon factorization of the binary expansion of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Nov 10 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).

Also the length of the Lyndon factorization of the inverted binary expansion of n, where the inverted digits are 1 minus the binary digits.

			The binary indices of 1..20 together with their co-Lyndon factorizations are:
   1:     (1) = (1)
   2:    (10) = (10)
   3:    (11) = (1)(1)
   4:   (100) = (100)
   5:   (101) = (10)(1)
   6:   (110) = (110)
   7:   (111) = (1)(1)(1)
   8:  (1000) = (1000)
   9:  (1001) = (100)(1)
  10:  (1010) = (10)(10)
  11:  (1011) = (10)(1)(1)
  12:  (1100) = (1100)
  13:  (1101) = (110)(1)
  14:  (1110) = (1110)
  15:  (1111) = (1)(1)(1)(1)
  16: (10000) = (10000)
  17: (10001) = (1000)(1)
  18: (10010) = (100)(10)
  19: (10011) = (100)(1)(1)
  20: (10100) = (10100)

The non-"co" version is A211100.
Positions of 1's are A275692.
The reversed version is A329326.
Cf. A000031, A001037, A059966, A060223, A211097, A296372, A296658, A329131, A329314, A329318, A329324, A329325.

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[Length[colynfac[IntegerDigits[n,2]]],{n,100}]

A329313 Length of the Lyndon factorization of the reversed binary expansion of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Nov 11 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. 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 sequence of reversed binary expansions of the nonnegative integers together with their Lyndon factorizations begins:
   0:      () = ()
   1:     (1) = (1)
   2:    (01) = (01)
   3:    (11) = (1)(1)
   4:   (001) = (001)
   5:   (101) = (1)(01)
   6:   (011) = (011)
   7:   (111) = (1)(1)(1)
   8:  (0001) = (0001)
   9:  (1001) = (1)(001)
  10:  (0101) = (01)(01)
  11:  (1101) = (1)(1)(01)
  12:  (0011) = (0011)
  13:  (1011) = (1)(011)
  14:  (0111) = (0111)
  15:  (1111) = (1)(1)(1)(1)
  16: (00001) = (00001)
  17: (10001) = (1)(0001)
  18: (01001) = (01)(001)
  19: (11001) = (1)(1)(001)
  20: (00101) = (00101)

The non-reversed version is A211100.
Positions of 1's are A328596.
The "co" version is A329326.
Binary Lyndon words are counted by A001037 and ranked by A102659.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose reversed binary expansion is a aperiodic are A328594.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A000031, A027375, A059966, A060223, A121016, A211097, A275692, A329131, A329314,  A329317, A329325.

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]]&]]]];
Table[If[n==0,0,Length[lynfac[Reverse[IntegerDigits[n,2]]]]],{n,0,30}]

A329326 Length of the co-Lyndon factorization of the reversed binary expansion of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 4, 3, 4, 4, 5, 2, 3, 2, 4, 3, 3, 2, 5, 3, 4, 3, 5, 4, 5, 5, 6, 2, 3, 2, 4, 2, 3, 2, 5, 3, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 5, 4, 4, 3, 6, 4, 5, 4, 6, 5, 6, 6, 7, 2, 3, 2, 4, 2, 3, 2, 5, 3, 3, 2, 4, 3, 3, 2, 6, 3, 4, 2, 5, 4, 3, 2

Offset: 1

Gus Wiseman, Nov 11 2019

nonn

First differs from A211100 at a(77) = 3, A211100(77) = 2. The reversed binary expansion of 77 is (1011001), with co-Lyndon factorization (10)(1100)(1), while the binary expansion is (1001101), with Lyndon factorization of (1)(001101).

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 certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

			The reversed binary expansion of each positive integer together with their co-Lyndon factorizations begins:
   1:     (1) = (1)
   2:    (01) = (0)(1)
   3:    (11) = (1)(1)
   4:   (001) = (0)(0)(1)
   5:   (101) = (10)(1)
   6:   (011) = (0)(1)(1)
   7:   (111) = (1)(1)(1)
   8:  (0001) = (0)(0)(0)(1)
   9:  (1001) = (100)(1)
  10:  (0101) = (0)(10)(1)
  11:  (1101) = (110)(1)
  12:  (0011) = (0)(0)(1)(1)
  13:  (1011) = (10)(1)(1)
  14:  (0111) = (0)(1)(1)(1)
  15:  (1111) = (1)(1)(1)(1)
  16: (00001) = (0)(0)(0)(0)(1)
  17: (10001) = (1000)(1)
  18: (01001) = (0)(100)(1)
  19: (11001) = (1100)(1)
  20: (00101) = (0)(0)(10)(1)

The non-"co" version is A211100.
Positions of 2's are A329357.
Numbers whose binary expansion is co-Lyndon are A275692.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A000031, A001037, A059966, A060223, A211097, A296372, A296658, A328596, A329131, A329314, A329318, A329324, A329325.

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[Length[colynfac[Reverse[IntegerDigits[n,2]]]],{n,100}]

A281013 Tetrangle T(n,k,i) = i-th part of k-th prime composition of n.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 1, 1, 3, 1, 4, 2, 1, 1, 1, 2, 2, 1, 3, 1, 1, 3, 2, 4, 1, 5, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 3, 1, 2, 3, 2, 1, 4, 1, 1, 4, 2, 5, 1, 6, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 1, 3, 2, 2, 3, 3, 1, 4, 1, 1, 1, 4, 1, 2, 4, 2, 1, 4, 3, 5, 1, 1, 5, 2, 6, 1, 7

Offset: 1

Gus Wiseman, Jan 12 2017

The *-product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling them together. Every finite positive integer sequence has a unique *-factorization using prime compositions P = {(1), (2), (21), (3), (211), ...}. See A060223 and A228369 for details.

These are co-Lyndon compositions, ordered first by sum and then lexicographically. - Gus Wiseman, Nov 15 2019

			The prime factorization of (1, 1, 4, 2, 3, 1, 5, 5) is: (11423155) = (1)*(1)*(5)*(5)*(4231). The prime factorizations of the initial terms of A000002 are:
             (1) = (1)
            (12) = (1)*(2)
           (122) = (1)*(2)*(2)
          (1221) = (1)*(221)
         (12211) = (1)*(2211)
        (122112) = (1)*(2)*(2211)
       (1221121) = (1)*(221121)
      (12211212) = (1)*(2)*(221121)
     (122112122) = (1)*(2)*(2)*(221121)
    (1221121221) = (1)*(221)*(221121)
   (12211212212) = (1)*(2)*(221)*(221121)
  (122112122122) = (1)*(2)*(2)*(221)*(221121).
Read as a sequence:
(1), (2), (21), (3), (211), (31), (4), (2111), (221), (311), (32), (41), (5).
Read as a triangle:
(1)
(2)
(21), (3)
(211), (31), (4)
(2111), (221), (311), (32), (41), (5).
Read as a sequence of triangles:
1    2    2 1    2 1 1    2 1 1 1    2 1 1 1 1    2 1 1 1 1 1
          3      3 1      2 2 1      2 2 1 1      2 1 2 1 1
                 4        3 1 1      3 1 1 1      2 2 1 1 1
                          3 2        3 1 2        2 2 2 1
                          4 1        3 2 1        3 1 1 1 1
                          5          4 1 1        3 1 1 2
                                     4 2          3 1 2 1
                                     5 1          3 2 1 1
                                     6            3 2 2
                                                  3 3 1
                                                  4 1 1 1
                                                  4 1 2
                                                  4 2 1
                                                  4 3
                                                  5 1 1
                                                  5 2
                                                  6 1
                                                  7.

Cf. A000740, A215474, A228369, A277427.
The binary version is A329318.
The binary non-"co" version is A102659.
A sequence listing all Lyndon compositions is A294859.
Numbers whose binary expansion is co-Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Binary Lyndon words are A001037.
Lyndon compositions are A059966.
Normal Lyndon words are A060223.
Cf. A211097, A211100, A296372, A296373, A298941, A329131, A329312, A329313, A329314, A329324, A329326.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n],colynQ],lexsort],{n,5}] (* Gus Wiseman, Nov 15 2019 *)

Row lengths are A059966(n) = number of prime compositions of n.

A329314 Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 4, 1, 2, 1, 1, 1, 2, 2, 1, 3, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, Nov 11 2019

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).

			Triangle begins:
   0: ()         20: (1211)      40: (12111)     60: (111111)
   1: (1)        21: (122)       41: (123)       61: (11112)
   2: (11)       22: (131)       42: (1221)      62: (111111)
   3: (11)       23: (14)        43: (15)        63: (111111)
   4: (111)      24: (11111)     44: (1311)      64: (1111111)
   5: (12)       25: (113)       45: (132)       65: (16)
   6: (111)      26: (1121)      46: (141)       66: (151)
   7: (111)      27: (113)       47: (15)        67: (16)
   8: (1111)     28: (11111)     48: (111111)    68: (1411)
   9: (13)       29: (1112)      49: (114)       69: (16)
  10: (121)      30: (11111)     50: (1131)      70: (151)
  11: (13)       31: (11111)     51: (114)       71: (16)
  12: (1111)     32: (111111)    52: (11211)     72: (13111)
  13: (112)      33: (15)        53: (1122)      73: (133)
  14: (1111)     34: (141)       54: (1131)      74: (151)
  15: (1111)     35: (15)        55: (114)       75: (16)
  16: (11111)    36: (1311)      56: (111111)    76: (1411)
  17: (14)       37: (15)        57: (1113)      77: (16)
  18: (131)      38: (141)       58: (11121)     78: (151)
  19: (14)       39: (15)        59: (1113)      79: (16)

Row lengths are A211100.
Row sums are A029837, or, if the first term is 1, A070939.
Ignoring the first digit gives A329325.
Positions of rows of length 2 are A329327.
Binary Lyndon words are counted by A001037 and ranked by A102659.
Numbers whose reversed binary expansion is a Lyndon word are A328596.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A059966, A211097, A275692, A296372, A329313, A329315, A329318.

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]]&]]]];
Table[Length/@lynfac[If[n==0,{},IntegerDigits[n,2]]],{n,0,50}]

A329325 Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n with first digit removed.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 3, 1, 4, 2, 1, 1, 2, 2, 3, 1, 4, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 4, 1, 5, 3, 1, 1, 5, 4, 1, 5, 2, 1

Offset: 1

Gus Wiseman, Nov 11 2019

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).

			Triangle begins:
   1: ()        21: (22)       41: (23)       61: (1112)
   2: (1)       22: (31)       42: (221)      62: (11111)
   3: (1)       23: (4)        43: (5)        63: (11111)
   4: (11)      24: (1111)     44: (311)      64: (111111)
   5: (2)       25: (13)       45: (32)       65: (6)
   6: (11)      26: (121)      46: (41)       66: (51)
   7: (11)      27: (13)       47: (5)        67: (6)
   8: (111)     28: (1111)     48: (11111)    68: (411)
   9: (3)       29: (112)      49: (14)       69: (6)
  10: (21)      30: (1111)     50: (131)      70: (51)
  11: (3)       31: (1111)     51: (14)       71: (6)
  12: (111)     32: (11111)    52: (1211)     72: (3111)
  13: (12)      33: (5)        53: (122)      73: (33)
  14: (111)     34: (41)       54: (131)      74: (51)
  15: (111)     35: (5)        55: (14)       75: (6)
  16: (1111)    36: (311)      56: (11111)    76: (411)
  17: (4)       37: (5)        57: (113)      77: (6)
  18: (31)      38: (41)       58: (1121)     78: (51)
  19: (4)       39: (5)        59: (113)      79: (6)
  20: (211)     40: (2111)     60: (11111)    80: (21111)
For example, the trimmed binary expansion of 41 is (01001), with Lyndon factorization (01)(001), so row 41 is {2,3}.

Row lengths are A211097.
Row sums are A000523.
Keeping the first digit gives A329314.
Positions of singleton rows are A329327.
Binary Lyndon words are counted by A001037 and ranked by A102659.
Numbers whose reversed binary expansion is a Lyndon word are A328596.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A059966, A211097, A257250, A275692, A296372, A328594, A329313, A329315, A329318.

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]]&]]]];
Table[Length/@lynfac[Rest[IntegerDigits[n,2]]],{n,100}]

A334029 Length of the co-Lyndon factorization of the k-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, 1, 1, 2, 2, 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, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 2

Offset: 0

Gus Wiseman, Apr 14 2020

nonn

We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). 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, (1,0,0,1) has co-Lyndon factorization {(1),(1,0,0)}.

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.

			The 441st composition in standard order is (1,2,1,1,3,1), with co-Lyndon factorization {(1),(3,1),(2,1,1)}, so a(441) = 3.

The dual version is A329312.
The version for binary expansion is (also) A329312.
The version for reversed binary expansion is A329326.
Binary Lyndon/co-Lyndon words are counted by A001037.
Necklaces covering an initial interval are A019536.
Lyndon/co-Lyndon compositions are counted by A059966
Length of Lyndon factorization of binomial expansion is A211100.
Numbers whose prime signature is a necklace are A329138.
Length of Lyndon factorization of reversed binary expansion is A329313.
A list of all binary co-Lyndon words is A329318.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- 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.
- Reversed co-necklaces are A328595.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
- Reversed necklaces are A333943.
- Co-necklaces are A334028.
Cf. A034691, A060223, A102659, A211097, A292884, A296372, A328596, A329358, A329359, A329362, A329400, A329401, A333939.

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

A211099 Largest (i.e., leftmost) Lyndon word in Lyndon factorization of binary vectors of lengths 1, 2, 3, ... (written using 1's and 2's rather than 0's and 1's, since numbers > 0 in the OEIS cannot begin with 0).

Original entry on oeis.org

1, 2, 1, 12, 2, 2, 1, 112, 12, 122, 2, 2, 2, 2, 1, 1112, 112, 1122, 12, 12, 122, 1222, 2, 2, 2, 2, 2, 2, 2, 2, 1, 11112, 1112, 11122, 112, 11212, 1122, 11222, 12, 12, 12, 12122, 122, 122, 1222, 12222, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 111112, 11112, 111122, 1112, 111212, 11122, 111222, 112, 112, 11212, 112122, 1122, 112212, 11222, 112222, 12

Offset: 1

N. J. A. Sloane, Apr 01 2012

nonn

Any binary word has a unique factorization as a product of nonincreasing Lyndon words (see Lothaire). Here we look at the Lyndon factorizations of the binary vectors 0,1, 00,01,10,11, 000,001,010,011,100,101,110,111, 0000,...
See A211097, A211099, A211100 for further information, including Maple code.
The smallest (or rightmost) factors are given by A211095 and A211096, offset by 2.

			Here are the Lyndon factorizations of the first few binary vectors:
.0.
.1.
.0.0.
.01.
.1.0.
.1.1.
.0.0.0.
.001.
.01.0.
.011.
.1.0.0.
.1.01.
.1.1.0.
.1.1.1.
.0.0.0.0.
...
The real sequence (written with 0's and 1's rather than 1's and 2's) is:
0, 1, 0, 01, 1, 1, 0, 001, 01, 011, 1, 1, 1, 1, 0, 0001, 001, 0011, 01, 01, 011, 0111, 1, 1, 1, 1, 1, 1, 1, 1, 0, 00001, 0001, 00011, 001, 00101, 0011, 00111, 01, 01, 01, 01011, 011, 011, 0111, 01111, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 000001, 00001, ...

M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. See Theorem 5.1.5, p. 67.
G. Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42

N. J. A. Sloane, Maple programs for A211097 etc.

Cf. A001037 (number of Lyndon words of length m); A102659 (list thereof), A211100.

Cf. A211095-A211099.

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

cp's OEIS Frontend

A211100 Number of factors in Lyndon factorization of binary expansion of n.

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A329312 Length of the co-Lyndon factorization of the binary expansion of n.

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A329313 Length of the Lyndon factorization of the reversed binary expansion of n.

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A329326 Length of the co-Lyndon factorization of the reversed binary expansion of n.

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A281013 Tetrangle T(n,k,i) = i-th part of k-th prime composition of n.

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A329314 Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n.

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A329325 Irregular triangle read by rows where row n gives the lengths of the components in the Lyndon factorization of the binary expansion of n with first digit removed.

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A334029 Length of the co-Lyndon factorization of the k-th composition in standard order.

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A211099 Largest (i.e., leftmost) Lyndon word in Lyndon factorization of binary vectors of lengths 1, 2, 3, ... (written using 1's and 2's rather than 0's and 1's, since numbers > 0 in the OEIS cannot begin with 0).

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