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| In [[combinatorics]], a '''square-free word''' is a [[String (computer science)|word]] that does not contain any [[Substring#Substring|subword]] twice in a row.
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| A square-free word is thus one that [[Avoidable pattern|avoids the pattern]] ''XX''.<ref name=LotII112>Lothaire (2011) p.112</ref><ref name=LotII114>Lothaire (2011) p.114</ref>
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| ==Examples==
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| 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,<ref name=LotII113/> as proved by [[Axel Thue]].<ref>A. Thue, Über unendliche Zeichenreihen, Norske Vid. Skrifter I Mat.-Nat. Kl., Christiania 7 (1906) 1–22.</ref><ref>A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Skrifter I Mat.-Nat. Kl., Christiania 1 (1912) 1–67.</ref>
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| 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]].<ref name=PF104>Pytheas Fogg (2002) p.104</ref><ref name=BLRS97>Berstel et al (2009) p.97</ref>
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| An example found by [[John Leech (mathematician)|John Leech]]<ref>{{cite journal | first=J. | last=Leech | authorlink=John Leech (mathematician) | title=A problem on strings of beads | journal=Math. Gazette | volume=41 | year=1957 | pages=277–278 | zbl=0079.01101 }}</ref> is defined recursively over the alphabet {''a, b, c''}. Let <math>w_1</math> be any word starting with the letter ''a''. Define the words <math> \{w_i \mid i \in \mathbb{N} \}</math> recursively as follows: the word <math>w_{i+1}</math> is obtained from <math>w_i</math> by replacing each ''a'' in <math>w_i</math> 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
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| :''abcbacbcabcbabcacbacabcacbcabacbabcabacbcacbacabcacb''...
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| ==Related concepts==
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| The Thue–Morse sequence is an example of a cube-free word over a binary alphabet.<ref name=LotII113>Lothaire (2011) p.113</ref> This sequence is not square-free but is "almost" so: the [[Critical exponent of a word|critical exponent]] is 2.<ref>{{cite book | title=Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, CA, USA, June 26-29, 2006 | volume=4036 | series=Lecture Notes in Computer Science | editor1-first=Oscar H. | editor1-last=Ibarra | editor2-first=Zhe | editor2-last=Dang | publisher=[[Springer-Verlag]] | year=2006 | isbn=3-540-35428-X | first=Dalia | last=Krieger | chapter=On critical exponents in fixed points of non-erasing morphisms | pages=280-291 | zbl=1227.68074 }}</ref> The Thue–Morse sequence has no '''overlap''' or ''overlapping square'', instances of 0''X''0''X''0 or 1''X''1''X''1:<ref name=LotII113/> it is essentially the only infinite binary word with this property.<ref name=BLRS81>Berstel et al (2009) p.81</ref>
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| The [[Thue number]] of a [[Graph (mathematics)|graph]] ''G'' is the smallest number ''k'' such that ''G'' has a ''k''-coloring for which the sequence of colors along every non-repeating path is squarefree.
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| == Notes ==
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| {{reflist}}
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| == References ==
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| * {{cite book | last1=Berstel | first1=Jean | last2=Lauve | first2=Aaron | last3=Reutenauer | first3=Christophe | last4=Saliola | first4=Franco V. | title=Combinatorics on words. Christoffel words and repetitions in words | series=CRM Monograph Series | volume=27 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2009 | isbn=978-0-8218-4480-9 | url=http://www.ams.org/bookpages/crmm-27 | zbl=1161.68043 }}
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| * {{cite book | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Combinatorics on words | publisher=[[Cambridge University Press]] | location=Cambridge | year= 1997 | isbn= 0-521-59924-5 }}.
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| * {{cite book | last=Lothaire | first=M. | authorlink=M. Lothaire | title=Algebraic combinatorics on words | others=With preface by Jean Berstel and Dominique Perrin | edition=Reprint of the 2002 hardback | series=Encyclopedia of Mathematics and Its Applications | volume=90| publisher=[[Cambridge University Press]] | year=2011 | isbn=978-0-521-18071-9 | zbl=1221.68183 }}
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| * {{cite book | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}
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| [[Category:Formal languages]]
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| [[Category:Combinatorics on words]]
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