# Crucial words for abelian powers

Glen, A., Halldórsson, B. and Kitaev, S.
(2009)
*Crucial words for abelian powers.*
In: Diekert, V. and Nowotka, D., (eds.)
Developments in Language Theory, Lecture Notes in Computer Science, Vol. 5583.
Springer, pp. 264-275.

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## Abstract

Let k ≥ 2 be an integer. An abelian k -th power is a word of the form X 1 X 2 ⋯ X k where X i is a permutation of X 1 for 2 ≤ i ≤ k. In this paper, we consider crucial words for abelian k-th powers, i.e., finite words that avoid abelian k-th powers, but which cannot be extended to the right by any letter of their own alphabets without creating an abelian k-th power. More specifically, we consider the problem of determining the minimal length of a crucial word avoiding abelian k-th powers. This problem has already been solved for abelian squares by Evdokimov and Kitaev [6], who showed that a minimal crucial word over an n-letter alphabet An={1,2,…,n} avoiding abelian squares has length 4n − 7 for n ≥ 3. Extending this result, we prove that a minimal crucial word over An avoiding abelian cubes has length 9n − 13 for n ≥ 5, and it has length 2, 5, 11, and 20 for n = 1,2,3, and 4, respectively. Moreover, for n ≥ 4 and k ≥ 2, we give a construction of length k 2(n − 1) − k − 1 of a crucial word over An avoiding abelian k-th powers. This construction gives the minimal length for k = 2 and k = 3.

Publication Type: | Book Chapter |
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Murdoch Affiliation: | School of Chemical and Mathematical Science |

Publisher: | Springer |

Copyright: | © 2009 Springer |

Notes: | 13th International Conference, DLT 2009, Stuttgart, Germany, June 30--July 3, 2009 |

URI: | http://researchrepository.murdoch.edu.au/id/eprint/19775 |

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