Square-free word
In combinatorics, a square-free word is a word (a sequence of characters) that does not contain any subword twice in a row.
Thus a square-free word is one that avoids the pattern XX.[1][2]
Examples
Over a two-letter alphabet {a, b} the only square-free words are the empty word and a, b, ab, ba, aba, and bab. However, there exist infinite square-free words in any alphabet with three or more symbols,[3] as proved by Axel Thue.[4][5]
One example of an infinite square-free word over an alphabet of size 3 is the word over the alphabet {0,±1} obtained by taking the first difference of the Thue–Morse sequence.[6][7] That is, from the Thue–Morse sequence
- 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...
one forms a new sequence in which each term is the difference of two consecutive terms of the Thue–Morse sequence. The resulting square-free word is
Another example found by John Leech[8] is defined recursively over the alphabet {a, b, c}. Let be any word starting with the letter a. Define the words recursively as follows: the word is obtained from by replacing each a in with abcbacbcabcba, each b with bcacbacabcacb, and each c with cabacbabcabac. It is possible to check that the sequence converges to the infinite square-free word
- abcbacbcabcbabcacbacabcacbcabacbabcabacbcacbacabcacb...
Related concepts
A cube-free word is one with no occurrence of www for a factor w. The Thue-Morse sequence is an example of a cube-free word over a binary alphabet.[3] This sequence is not square-free but is "almost" so: the critical exponent is 2.[9] The Thue–Morse sequence has no overlap or overlapping square, instances of 0X0X0 or 1X1X1:[3] it is essentially the only infinite binary word with this property.[10]
The Thue number of a graph G is the smallest number k such that G has a k-coloring for which the sequence of colors along every simple path is squarefree.
The Kolakoski sequence is an example of a cube-free sequence.
An abelian p-th power is a subsequence of the form where each is a permutation of . There is no abelian-square-free infinite word over an alphabet of size three: indeed, every word of length eight over such an alphabet contains an abelian square. There is an infinite abelian-square-free word over an alphabet of size five.[11]
Notes
- ↑ Lothaire (2011) p.112
- ↑ Lothaire (2011) p.114
- 1 2 3 Lothaire (2011) p.113
- ↑ A. Thue, Über unendliche Zeichenreihen, Norske Vid. Skrifter I Mat.-Nat. Kl., Christiania 7 (1906) 1–22.
- ↑ A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Skrifter I Mat.-Nat. Kl., Christiania 1 (1912) 1–67.
- ↑ Pytheas Fogg (2002) p.104
- ↑ Berstel et al (2009) p.97
- ↑ Leech, J. (1957). "A problem on strings of beads". Math. Gazette. 41: 277–278. Zbl 0079.01101.
- ↑ Krieger, Dalia (2006). "On critical exponents in fixed points of non-erasing morphisms". In Ibarra, Oscar H.; Dang, Zhe. Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, CA, USA, June 26-29, 2006. Lecture Notes in Computer Science. 4036. Springer-Verlag. pp. 280–291. ISBN 3-540-35428-X. Zbl 1227.68074.
- ↑ Berstel et al (2009) p.81
- ↑ Blanchet-Sadri, Francine; Simmons, Sean (2011). "Avoiding Abelian Powers in Partial Words". In Mauri, Giancarlo; Leporati, Alberto. Developments in Language Theory. Proceedings, 15th International Conference, DLT 2011, Milan, Italy, July 19-22, 2011. Lecture Notes in Computer Science. 6795. Berlin, Heidelberg: Springer-Verlag. pp. 70–81. doi:10.1007/978-3-642-22321-1_7. ISBN 978-3-642-22320-4. ISSN 0302-9743.
References
- Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. 27. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4480-9. Zbl 1161.68043.
- Lothaire, M. (1997). Combinatorics on words. Cambridge: Cambridge University Press. ISBN 0-521-59924-5..
- Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9. Zbl 1221.68183.
- Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. 1794. Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.