Abelian Pattern Avoidance in Partial Words

Abstract

Pattern avoidance is an important topic in combinatorics on words which dates back to the beginning of the twentieth century when Thue constructed an infinite word over a ternary alphabet that avoids squares, i.e., a word with no two adjacent identical factors. This result finds applications in various algebraic contexts where more general patterns than squares are considered. On the other hand, Erdős raised the question as to whether there exists an infinite word that avoids abelian squares, i.e., a word with no two adjacent factors being permutations of one another. Although this question was answered affirmately years later, knowledge of abelian pattern avoidance is rather limited. Recently, (abelian) pattern avoidance was initiated in the more general framework of partial words, which allow for undefined positions called holes. Here, we investigate conditions for a pattern to be abelian avoidable by a partial word with finitely or infinitely many holes.

This material is based upon work supported by the National Science Foundation under Grant Nos. DMS–0754154 and DMS–1060775. The Department of Defense is gratefully acknowledged. The authors thank Dimin Xu for valuable comments and suggestions. A research assignment from the University of North Carolina at Greensboro for the first author is gratefully acknowledged. Some of this assignment was spent at the LIAFA: Laboratoire d’Informatique Algorithmique: Fondements et Applications of Université Paris 7-Denis Diderot, Paris, France.