# Crucial abelian k-power-free words

Glen, Amy and Halldorsson, Bjarni and Kitaev, Sergey
(2010)
*Crucial abelian k-power-free words.*
Discrete Mathematics and Theoretical Computer Science, 12 (5).
pp. 83-96.

## 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: | Article |
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ID code: | 34650 |

Keywords: | abelian squares, abelian k-th power, crucial word, Electronic computers. Computer science, Discrete Mathematics and Combinatorics, Theoretical Computer Science, Computer Science(all) |

Subjects: | Science > Mathematics > Electronic computers. Computer science |

Department: | Faculty of Science > Computer and Information Sciences |

Depositing user: | Pure Administrator |

Date deposited: | 25 Oct 2011 10:15 |

Last modified: | 06 Jan 2017 10:11 |

Related URLs: | |

URI: | https://strathprints.strath.ac.uk/id/eprint/34650 |

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