# An abelian periodicity lemma

Simpson, J.
(2015)
*An abelian periodicity lemma.*
Theoretical Computer Science, 656
(B).
pp. 249-255.

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## Abstract

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.

Item Type: | Journal Article |
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Murdoch Affiliation: | School of Engineering and Information Technology |

Publisher: | Elsevier BV |

Copyright: | © 2016 Published by Elsevier B.V. |

URI: | http://researchrepository.murdoch.edu.au/id/eprint/29575 |

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