The avoidability of cubes under permutations

  • Authors:

  • Florin Manea;Mike Müller;Dirk Nowotka

  • Affiliations:

  • Institut für Informatik, Christian-Albrechts-Universität zu Kiel, Kiel, Germany;Institut für Informatik, Christian-Albrechts-Universität zu Kiel, Kiel, Germany;Institut für Informatik, Christian-Albrechts-Universität zu Kiel, Kiel, Germany

  • Venue:
  • DLT'12 Proceedings of the 16th international conference on Developments in Language Theory
  • Year:
  • 2012

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Abstract

In this paper we consider the avoidance of patterns in infinite words. Generalising the traditional problem setting, functional dependencies between pattern variables are allowed here, in particular, patterns involving permutations. One of the remarkable facts is that in this setting the notion of avoidability index (the smallest alphabet size for which a pattern is avoidable) is meaningless since a pattern with permutations that is avoidable in one alphabet can be unavoidable in a larger alphabet. We characterise the (un-)avoidability of all patterns of the form πi(x) πj(x) πk(x), called cubes under permutations here, for all alphabet sizes in both the morphic and antimorphic case.