Murdoch University Research Repository

## Welcome to the Murdoch University Research Repository

The Murdoch University Research Repository is an open access digital collection of research
created by Murdoch University staff, researchers and postgraduate students.

Learn more

# Crucial abelian k-power-free words

Glen, A.ORCID: 0000-0002-9434-3412, Halldórsson, B. and Kitaev, S. (2010) Crucial abelian k-power-free words. Discrete Mathematics & Theoretical Computer Science, 12 (5). pp. 83-96.

 Preview
PDF - Published Version
Download (108kB)
Free to read: http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/art...
*No subscription required

## Abstract

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.

Item Type: Journal Article Discrete Mathematics and Theoretical Computer Science © Discrete Mathematics and Theoretical Computer Science http://researchrepository.murdoch.edu.au/id/eprint/3883
 Item Control Page

## Downloads

Downloads per month over past year