Crucial Words for Abelian Powers

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 ${\mathcal{A}}_n = \{1,2,\ldots, n\}$ avoiding abelian squares has length 4n ? 7 for n ? 3. Extending this result, we prove that a minimal crucial word over ${\mathcal{A}}_n$ 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 ${\mathcal{A}}_n$ avoiding abelian k-th powers. This construction gives the minimal length for k = 2 and k = 3.