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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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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
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
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