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

Published online by Cambridge University Press:  25 April 2013

Francine Blanchet-Sadri ,
Sean Simmons ,
Amelia Tebbe  and
Amy Veprauskas
Francine Blanchet-Sadri
Affiliation:
Department of Computer Science, University of North Carolina, P.O. Box 26170, Greensboro, NC 27402–6170, USA.. [email protected]
Sean Simmons
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Building 2, Room 236, 77 Massachusetts Avenue, Cambridge, MA 02139–4307, USA
Amelia Tebbe
Affiliation:
Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA
Amy Veprauskas
Affiliation:
Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., P.O. Box 210089 Tucson, AZ 85721–0089, USA
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Abstract

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.

Keywords

Combinatorics on wordsFine and Wilf’s theorempartial wordsabelian periodsperiodsoptimal lengths
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 47 , Issue 3 , July 2013 , pp. 215 - 234
Copyright
© EDP Sciences 2013

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