Francine Blanchet-Sadri and Sean Simmons

**Abstract.** 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 *p*th 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 *p*th 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 *p*th powers under arbitrary insertion of holes in an infinite word.