Crucial words for abelian powers

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 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 avoiding abelian k-th powers. This construction gives the minimal length for k=2 and k=3. © 2009 Springer Berlin Heidelberg.