A210584 -id:A210584 - cp's OEIS frontend

A102659 List of Lyndon words on {1,2} sorted first by length and then lexicographically.

1, 2, 12, 112, 122, 1112, 1122, 1222, 11112, 11122, 11212, 11222, 12122, 12222, 111112, 111122, 111212, 111222, 112122, 112212, 112222, 121222, 122222, 1111112, 1111122, 1111212, 1111222, 1112112, 1112122, 1112212, 1112222, 1121122

Offset: 1

N. J. A. Sloane, Feb 03 2005

A Lyndon word is primitive (not a power of another word) and is earlier in lexicographic order than any of its cyclic shifts.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Bassino, J. Clement and C. Nicaud, The standard factorization of Lyndon words: an average point of view, Discrete Math. 290 (2005), 1-25.
Émilie Charlier, Manon Philibert, Manon Stipulanti, Nyldon words, arXiv:1804.09735 [math.CO], 2018. See Table 1.
A. M. Uludag, A. Zeytin and M. Durmus, Binary Quadratic Forms as Dessins, 2012. - From _N. J. A. Sloane_, Dec 31 2012
Wikipedia, Lyndon word
Reinhard Zumkeller, Haskell programs for some sequences concerning Lyndon words
Index entries for sequences related to Lyndon words

Cf. A001037, A074650, A102660, A210584, A210585.
The "co" version is A329318.
A triangular version is A296657.
A sequence listing all Lyndon compositions is A294859.
Numbers whose binary expansion is Lyndon are A328596.
Length of the Lyndon factorization of the binary expansion is A211100.
Cf. A059966, A060223, A275692, A281013, A296373, A329131, A329313.

Haskell
```
cf. link.
```

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
Join@@Table[FromDigits/@Select[Tuples[{1,2},n],lynQ],{n,5}] (* Gus Wiseman, Nov 14 2019 *)

is_A102659(n)={ vecsort(d=digits(n))!=d&&for(i=1,#d-1, n>[1,10^(#d-i)]*divrem(n,10^i)&&return); fordiv(#d,L,L<#d && d==concat(Col(vector(#d/L,i,1)~*vecextract(d,2^L-1))~)&&return); !setminus(Set(d),[1,2])} \\ The last check is the least expensive one, but not useful if we test only numbers with digits {1,2}.
for(n=1,6,p=vector(n,i,10^(n-i))~;forvec(d=vector(n,i,[1,2]),is_A102659(m=d*p)&&print1(m","))) \\ One could use is_A102660 instead of is_A102659 here. - M. F. Hasler, Mar 08 2014

A102659 = A102660 intersect A007931 = A213969 intersect A239016. - M. F. Hasler, Mar 10 2014

More terms from Franklin T. Adams-Watters, Dec 14 2006

Definition improved by Reinhard Zumkeller, Mar 23 2012

A102660 List of Lyndon words on {1,2,3} sorted first by length and then lexicographically.

Original entry on oeis.org

1, 2, 3, 12, 13, 23, 112, 113, 122, 123, 132, 133, 223, 233, 1112, 1113, 1122, 1123, 1132, 1133, 1213, 1222, 1223, 1232, 1233, 1322, 1323, 1332, 1333, 2223, 2233, 2333, 11112, 11113, 11122, 11123, 11132, 11133, 11212, 11213, 11222, 11223, 11232

Offset: 1

N. J. A. Sloane, Feb 03 2005

A Lyndon word is primitive (not a power of another word) and is earlier in lexicographic order than any of its cyclic shifts.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Bassino, J. Clement and C. Nicaud, The standard factorization of Lyndon words: an average point of view, Discrete Math. 290 (2005), 1-25.
Reinhard Zumkeller, Haskell programs for some sequences concerning Lyndon words
Wikipedia, Lyndon word
Index entries for sequences related to Lyndon words

Cf. A074650, A001037, A102659, A210584, A210585.

Cf. A027376.

Haskell
```
cf. link.
```

is_A102660(n)=is_A239016(n)&&is_A239017(n)
for(n=1, 5, p=vector(n, i, 10^(n-i))~; forvec(d=vector(n, i, [1, 3]), is_A102660(m=d*p)&&print1(m", "))) \\ M. F. Hasler, Mar 09 2014

Equals A239016 intersect A239017. - M. F. Hasler, Mar 09 2014

More terms from John W. Layman, Jan 24 2006

Definition improved by Reinhard Zumkeller, Mar 23 2012

A239016 Numbers not larger than any rotation of their digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 44, 45, 46, 47, 48, 49, 55, 56, 57, 58, 59, 66, 67, 68, 69, 77, 78, 79, 88, 89, 99, 111, 112, 113, 114, 115, 116, 117, 118, 119, 122, 123, 124, 125, 126, 127, 128, 129, 132

Offset: 1

M. F. Hasler, Mar 08 2014

The numbers with nonincreasing digits, A009994, form a subsequence which first differs at a(73)=132 (not in A009994) from this one.
This sequence is a subsequence of A072544: numbers whose smallest decimal digit is also the initial digit. A072544(65)=121 is the first such number not in this sequence.
This criterion involving "rotation" is part of the characterization of Lyndon words, see e.g. A102659, A102660, A210584, A210585. All of these are subsequences of this sequence. For example, A102659 = A213969 intersect A239016.

			The number 10 is excluded from this sequence because its "rotation" 01 is smaller than the number itself.
The same is the case for any number whose first digit is not the smallest one: rotating a smaller digit to the front will always yield a smaller number, independently of the other digits. For this reason, all terms must be in A072544.
a(73)=132 is in the sequence because the nontrivial rotations of its digits are 321 and 213, both larger than 132.

is_A239016(n)=vecsort(d=digits(n))==d||!for(i=1,#d-1,n>[1,10^(#d-i)]*divrem(n,10^i)&&return)

def ok(n):
    s = str(n)
    if "".join(sorted(s)) == s: return True
    return all(n <= int(s[i:] + s[:i]) for i in range(1, len(s)))
print(list(filter(ok, range(133)))) # Michael S. Branicky, Aug 21 2021

A210585 List of Lyndon words on {1,...,8} sorted first by length and then lexicographically.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 12, 13, 14, 15, 16, 17, 18, 23, 24, 25, 26, 27, 28, 34, 35, 36, 37, 38, 45, 46, 47, 48, 56, 57, 58, 67, 68, 78, 112, 113, 114, 115, 116, 117, 118, 122, 123, 124, 125, 126, 127, 128, 132, 133, 134, 135, 136, 137, 138, 142, 143, 144

Offset: 1

Reinhard Zumkeller, Mar 23 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, Haskell programs for some sequences concerning Lyndon words
Wikipedia, Lyndon word
Index entries for sequences related to Lyndon words

Cf. A102659, A102660, A210584.

Haskell
```
cf. link.
```

cp's OEIS Frontend

A102659 List of Lyndon words on {1,2} sorted first by length and then lexicographically.

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A102660 List of Lyndon words on {1,2,3} sorted first by length and then lexicographically.

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A239016 Numbers not larger than any rotation of their digits.

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A210585 List of Lyndon words on {1,...,8} sorted first by length and then lexicographically.

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Haskell