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Crucial abelian k-power-free words.
- Source :
-
Discrete Mathematics & Theoretical Computer Science (DMTCS) . Dec2010, Vol. 12 Issue 5, p83-96. 14p. - Publication Year :
- 2010
-
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 ge; 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *ABELIAN equations
*PERMUTATIONS
*ABELIAN groups
*MATHEMATICS
*ALGEBRA
Subjects
Details
- Language :
- English
- ISSN :
- 13658050
- Volume :
- 12
- Issue :
- 5
- Database :
- Academic Search Index
- Journal :
- Discrete Mathematics & Theoretical Computer Science (DMTCS)
- Publication Type :
- Academic Journal
- Accession number :
- 60989022