Unary Patterns with Permutations

  • James Currie
  • Florin Manea
  • Dirk Nowotka
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9168)


Thue characterized completely the avoidability of unary patterns. Adding function variables gives a general setting capturing avoidance of powers, avoidance of patterns with palindromes, avoidance of powers under coding, and other questions of recent interest. Unary patterns with permutations have been previously analysed only for lengths up to 3. Consider a pattern \(p=\pi _{i_1}(x)\ldots \pi _{i_r}(x)\), with \(r\ge 4\), x a word variable over an alphabet \(\Sigma \) and \(\pi _{i_j}\) function variables, to be replaced by morphic or antimorphic permutations of \(\Sigma \). If \(|\Sigma |\ge 3\), we show the existence of an infinite word avoiding all pattern instances having \(|x|\ge 2\). If \(|\Sigma |=3\) and all \(\pi _{i_j}\) are powers of a single \(\pi \), the length restriction is removed. In general, the restriction on x cannot be removed, even for powers of permutations: for every positive integer n there exists N and a pattern \(\pi ^{i_1}(x)\ldots \pi ^{i_n}(x)\) which is unavoidable over all \(\Sigma \).


Function Variable Unary Pattern Negative Integer Chinese Remainder Theorem Pattern Instance 
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  1. 1.
    Bischoff, B., Currie, J., Nowotka, D.: Unary patterns with involution. Int. J. Found. Comput. Sci. 23(8), 1641–1652 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chiniforooshan, E., Kari, L., Xu, Z.: Pseudopower avoidance. Fundam. Inform. 114(1), 55–72 (2012)MathSciNetGoogle Scholar
  3. 3.
    Currie, J.: Pattern avoidance: themes and variations. Theoret. Comput. Sci. 339(1), 7–18 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hall, M.: Lectures on Modern Mathematics, vol. 2, chap. Generators and relations in groups - The Burnside problem, pp. 42–92. Wiley, New York (1964)Google Scholar
  5. 5.
    Lothaire, M.: Combinatorics on Words. Cambridge University Press (1997)Google Scholar
  6. 6.
    Manea, F., Müller, M., Nowotka, D.: The avoidability of cubes under permutations. In: Yen, H.-C., Ibarra, O.H. (eds.) DLT 2012. LNCS, vol. 7410, pp. 416–427. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  7. 7.
    Thue, A.: Über unendliche Zeichenreihen. Norske Vid. Skrifter I. Mat.-Nat. Kl., Christiania 7, 1–22 (1906)Google Scholar
  8. 8.
    Thue, A.: Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Skrifter I. Mat.-Nat. Kl., Christiania 1, 1–67 (1912)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of WinnipegWinnipegCanada
  2. 2.Department of Computer ScienceKiel UniversityKielGermany

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