{"pk":64963,"title":"On the number of squares in a finite word","subtitle":null,"abstract":"Let \\(u\\) be a nonempty finite word, a square is a word of the form \\(uu\\). In this paper, we prove that for a given finite word \\(w\\), the number of distinct square factors of \\(w\\) is bounded by \\(|w|-|\\operatorname{Alph}(w)|\\), where \\(|w|\\) denotes the length of \\(w\\) and \\(|\\operatorname{Alph}(w)|\\) denotes the number of distinct letters in \\(w\\). This result answers positively a conjecture stated by Fraenkel and Simpson in 1998 and the \\(d\\)-step conjecture stated by Deza, Franek and Jiang in 2011.\n \nMathematics Subject Classifications: 68R15, 68R10, 68R05\n \nKeywords: Combinatorics on words, squares, repetition","language":"en","license":{"name":"Creative Commons Attribution 4.0","short_name":"CC BY 4.0","text":"Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.","url":"https://creativecommons.org/licenses/by/4.0"},"keywords":[{"word":"Combinatorics on words"},{"word":"squares"},{"word":"repetition"}],"section":"Research Articles","is_remote":true,"remote_url":"https://escholarship.org/uc/item/8k6180qp","frozenauthors":[{"first_name":"Shuo","middle_name":"","last_name":"Li","name_suffix":"","institution":"LACIM, Université du Québec à Montréal, Montréal, Québec, Canada -- Departement of Mathematics and Statistics, The University of Winnipeg, Winnipeg, Manitoba, Canada","department":""}],"date_submitted":"2025-03-14T16:16:16Z","date_accepted":"2025-03-14T16:16:16Z","date_published":"2025-03-15T07:00:00Z","render_galley":null,"galleys":[{"label":"","type":"pdf","path":"https://journalpub.escholarship.org/combinatorial_theory/article/64963/galley/49773/download/"}]}