{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,1]],"date-time":"2026-02-01T19:50:08Z","timestamp":1769975408165,"version":"3.49.0"},"reference-count":26,"publisher":"EDP Sciences","issue":"3","license":[{"start":{"date-parts":[[2013,4,25]],"date-time":"2013-04-25T00:00:00Z","timestamp":1366848000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.edpsciences.org\/en\/authors\/copyright-and-licensing"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Theor. Inf. Appl."],"accepted":{"date-parts":[[2013,1,21]]},"published-print":{"date-parts":[[2013,7]]},"abstract":"<jats:p>Recently, Constantinescu and Ilie proved a variant of the well-known periodicity theorem\n          of Fine and Wilf in the case of two relatively prime abelian periods and conjectured a\n          result for the case of two non-relatively prime abelian periods. In this paper, we answer\n          some open problems they suggested. We show that their conjecture is false but we give\n          bounds, that depend on the two abelian periods, such that the conjecture is true for all\n          words having length at least those bounds and show that some of them are optimal. We also\n          extend their study to the context of partial words, giving optimal lengths and describing\n          an algorithm for constructing optimal words. <\/jats:p>","DOI":"10.1051\/ita\/2013034","type":"journal-article","created":{"date-parts":[[2013,4,25]],"date-time":"2013-04-25T07:24:20Z","timestamp":1366874660000},"page":"215-234","source":"Crossref","is-referenced-by-count":9,"title":["Abelian periods, partial words, and an extension of a theorem\n          of Fine and Wilf"],"prefix":"10.1051","volume":"47","author":[{"given":"Francine","family":"Blanchet-Sadri","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sean","family":"Simmons","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amelia","family":"Tebbe","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amy","family":"Veprauskas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"250","published-online":{"date-parts":[[2013,4,25]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1016\/j.dam.2009.11.010","volume":"158","author":"Avgustinovich","year":"2010","journal-title":"Discrete Appl. 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In JM 2010, 13i\u00e8mes Journ\u00e9es Montoises d\u2019Informatique Th\u00e9orique, Amiens, France (2010)."},{"key":"R10","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/S0304-3975(98)00251-5","volume":"218","author":"Castelli","year":"1999","journal-title":"Theoret. Comput. Sci."},{"key":"R11","doi-asserted-by":"crossref","unstructured":"C. Choffrut and J. Karhum\u00e4ki, Combinatorics of Words. In Handbook of Formal Languages, edited by G. Rozenberg and A. Salomaa, Springer-Verlag, Berlin Vol. 1 (1997) 329\u2013438.","DOI":"10.1007\/978-3-642-59136-5_6"},{"key":"R12","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.tcs.2005.01.007","volume":"339","author":"Constantinescu","year":"2005","journal-title":"Theor. Comput. Sci."},{"key":"R13","first-page":"167","volume":"89","author":"Constantinescu","year":"2006","journal-title":"Bull. Eur. Assoc. Theor. Comput. 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