### Abstract

Let k ≥ 2 be an integer. An abelian k -th power is a word of the form X 1 X 2 ⋯ 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 [6], 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 k 2(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.

Original language | English |
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Title of host publication | Developments in Language Theory |

Subtitle of host publication | 13th International Conference, DLT 2009, Stuttgart, Germany, June 30-July 3, 2009. Proceedings |

Editors | Volker Diekert, Dirk Nowotka |

Pages | 264-275 |

Number of pages | 12 |

DOIs | |

Publication status | Published - 2009 |

Event | 13th Conference on Developments in Language Theory - Stuttgart, Germany Duration: 30 Jun 2009 → 3 Jul 2009 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer |

Volume | 5583 |

ISSN (Print) | 0302-9743 |

### Conference

Conference | 13th Conference on Developments in Language Theory |
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Country | Germany |

City | Stuttgart |

Period | 30/06/09 → 3/07/09 |

### Keywords

- pattern avoidance
- abelian square-free word
- abelian cube-free word
- abelian power
- crucial word

## Cite this

Glen, A., Halldorsson, B., & Kitaev, S. (2009). Crucial words for abelian powers. In V. Diekert, & D. Nowotka (Eds.),

*Developments in Language Theory: 13th International Conference, DLT 2009, Stuttgart, Germany, June 30-July 3, 2009. Proceedings*(pp. 264-275). (Lecture Notes in Computer Science; Vol. 5583). https://doi.org/10.1007/978-3-642-02737-6_21