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fine and wilf’s theorem

RAIRO - Theoretical Informatics and Applications (1)

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Abelian periods, partial words, and an extension of a theoremof Fine and Wilf ∗
Francine Blanchet-Sadri, Sean Simmons, Amelia Tebbe, Amy Veprauskas
Journal:

RAIRO - Theoretical Informatics and Applications / Volume 47 / Issue 3 / July 2013

Published online by Cambridge University Press:

25 April 2013, pp. 215-234

Print publication:

July 2013
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Recently, Constantinescu and Ilie proved a variant of the well-known periodicity theoremof Fine and Wilf in the case of two relatively prime abelian periods and conjectured aresult for the case of two non-relatively prime abelian periods. In this paper, we answersome open problems they suggested. We show that their conjecture is false but we givebounds, that depend on the two abelian periods, such that the conjecture is true for allwords having length at least those bounds and show that some of them are optimal. We alsoextend their study to the context of partial words, giving optimal lengths and describingan algorithm for constructing optimal words.