Avoiding abelian squares in partial words

  • Authors:

  • F. Blanchet-Sadri;Jane I. Kim;Robert Mercaş;William Severa;Sean Simmons;Dimin Xu

  • Affiliations:

  • Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 27402-6170, USA;Department of Mathematics, Columbia University, 2960 Broadway, New York, NY 10027-6902, USA;GRLMC, Departament de Filologies Romíniques, Universitat Rovira i Virgili, Av. Catalunya 35, Tarragona, 43002, Spain;Department of Mathematics, University of Florida, 358 Little Hall, P.O. Box 118105, Gainesville, FL 32611-8105, USA;Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX 78712-0233, USA;Department of Mathematics, Bard College, 30 Campus Road, Annandale-on-Hudson, NY 12504, USA

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2012

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Abstract

Erdos raised the question whether there exist infinite abelian square-free words over a given alphabet, that is, words in which no two adjacent subwords are permutations of each other. It can easily be checked that no such word exists over a three-letter alphabet. However, infinite abelian square-free words have been constructed over alphabets of sizes as small as four. In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes. In particular, we give lower and upper bounds for the number of letters needed to construct infinite abelian square-free partial words with finitely or infinitely many holes. Several of our constructions are based on iterating morphisms. In the case of one hole, we prove that the minimal alphabet size is four, while in the case of more than one hole, we prove that it is five. We also investigate the number of partial words of length n with a fixed number of holes over a five-letter alphabet that avoid abelian squares and show that this number grows exponentially with n.