Crucial abelian k-power-free words

In 1961, Erdős asked whether or not there exist words of arbitrary length over a fixed finite alphabet that avoid patterns of the form XX' where X' is a permutation of X (called abelian squares). This problem has since been solved in the affirmative in a series of papers from 1968 to 1992. Much less is known in the case of abelian k-th powers, i.e., words of the form X1X2⋯Xk where Xi is a permutation of X1 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 (2004), 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 k2(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. For k ≥4 and n≥5, we provide a lower bound for the length of crucial words over An avoiding abelian k-th powers.