Crucial abelian k-power-free words

• Published in: Discrete mathematics and theoretical computer science Vol. 12 no. 5; no. Combinatorics; pp. 83 - 96
• Main Authors: Glen, Amy, Halldorsson, Bjarni V., Kitaev, Sergey
• Format: Journal Article
• Language: English
• Published: Nancy DMTCS 01.01.2010; Discrete Mathematics & Theoretical Computer Science
• Subjects: [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]; abelian cube-free word; abelian power; abelian square-free word; Computer Science; crucial word; Discrete Mathematics; Finite element analysis; pattern avoidance; Theoretical mathematics; zimin word; abelian cube-free word; crucial word; abelian power; Zimin word; pattern avoidance; abelian square-free word
• ISSN: 1365-8050; 1462-7264; 1365-8050
• DOI: 10.46298/dmtcs.513

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 <= 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 A(n) = \1, 2, ..., n\ avoiding abelian squares has length 4n - 7 for n >= 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.