Fine and Wilf words for any periods II

Relational Fine and Wilf words

We consider interaction properties of relational periods, where the relation is a compatibility relation on words induced by a relation on letters. By the famous theorem of Fine and Wilf, p + q − gcd(p, q) − 1 is the maximal length of a word having periods p and q but not period gcd(p, q). Such words of maximal length are called extremal Fine and Wilf words. In this paper we study properties of...

Fine and Wilf Words for Any Periods

Interaction Properties of Relational Periods

Similarity relations, i.e., compatibility relations on words induced by relations on letters were introduced as a generalization of partial words in [9](i). There and in [10, 11] the main focus was on the effect of these relations on coding properties and on the defect theorem of words. In the article [8] we started the study of the interaction properties of periods with respect to similarity r...

The theorem of Fine and Wilf for relational periods

We consider relational periods, where the relation is a compatibility relation on words induced by a relation on letters. We prove a variant of the theorem of Fine and Wilf for a (pure) period and a relational period.

Abelian periods, partial words, and an extension of a theorem of Fine and Wilf

Recently, Constantinescu and Ilie proved a variant of the wellknown periodicity theorem of Fine and Wilf in the case of two relatively prime abelian periods and conjectured a result for the case of two nonrelatively prime abelian periods. In this paper, we answer some open problems they suggested. We show that their conjecture is false but we give bounds, that depend on the two abelian periods,...