Abelian Periods on Partial Words

Abstract

Recently, Constantinescu and Ilie proved a variant of the well known periodicity theorem of Fine and Wilf in the case of two relatively prime abelian periods. More precisely, any full word having two such periods p, q and length at least 2pq-1 has also gcd(p,q)=1 as a period. In this paper, we answer some open problems they suggested by proving that their length is optimal, by giving a result for the case of two non-relatively prime abelian periods, and by studying the case of any number of abelian periods. We also extend their result in the context of partial words.

*Keywords*: Combinatorics on words; Fine and Wilf's theorem; partial words; abelian periods; periods; optimal lengths.