Crucial abelian k-power-free words
Amy Glen, Bjarni V. Halldórsson, Sergey Kitaev · Discrete Mathematics & Theoretical Computer Science · 2010
Combinatorics In 1961, Erdos 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 ... X-k where X-i is a permutation of X-1 for 2 = 3. Extending this result, we prove that a minimal crucial word over 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 A(n) 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 A(n) avoiding abelian k-th powers.