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An abelian periodicity lemma

Simpson, J. (2015) An abelian periodicity lemma. Theoretical Computer Science . In Press.

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We write x≺yx≺y when x and y are vectors with each element of x less than or equal to the corresponding element of y and P(w)P(w) for the Parikh vector of a word w. A word w has abelian period p if it has the form u0u1⋯umum+1u0u1⋯umum+1 with |u0|≤p|u0|≤p, |ui|=p|ui|=p for i=1,…mi=1,…m and |um+1|≤p|um+1|≤p, and P(u0)≺PP(u0)≺P, P(u0)=PP(u0)=P for i=1,…,mi=1,…,m and P(um+1)≺PP(um+1)≺P for some vector P. Blanchet-Sadri et al. conjectured that if a word has abelian periods pd and qd , where gcd⁡(p,q)=1gcd⁡(p,q)=1, and length at least 2pqd−12pqd−1 then the number of distinct letters appearing in the word is at most d , and that under certain conditions the bound may be reduced to 2pqd−22pqd−2. We prove their conjecture.

Publication Type: Journal Article
Murdoch Affiliation: School of Engineering and Information Technology
Publisher: Elsevier BV
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