Developments in Language Theory

Volume 6795 of the series Lecture Notes in Computer Science pp 70-81

Avoiding Abelian Powers in Partial Words

  • Francine Blanchet-SadriAffiliated withDepartment of Computer Science, University of North Carolina
  • , Sean SimmonsAffiliated withDepartment of Mathematics, The University of Texas at Austin

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We study abelian repetitions in partial words, or sequences that may contain some unknown positions or holes. First, we look at the avoidance of abelian pth powers in infinite partial words, where p > 2, extending recent results regarding the case where p = 2. We investigate, for a given p, the smallest alphabet size needed to construct an infinite partial word with finitely or infinitely many holes that avoids abelian pth powers. We construct in particular an infinite binary partial word with infinitely many holes that avoids 6th powers. Then we show, in a number of cases, that the number of abelian p-free partial words of length n with h holes over a given alphabet grows exponentially as n increases. Finally, we prove that we cannot avoid abelian pth powers under arbitrary insertion of holes in an infinite word.