Crucial Words for Abelian Powers

A word is crucial with respect to a given set of prohibited words (or simply prohibitions) if it avoids the prohibitions but it cannot be extended to the right by any letter of its alphabet without creating a prohibition. A minimal crucial word is a crucial word of the shortest length. A word W contains an abelian k-th power if W has a factor of the form X1X2 . . . Xk where Xi is a permutation of X1 for 2 i k. When k = 2 or 3, one deals with abelian squares and abelian cubes, respectively. Evdokimov and Kitaev [6] have shown that a minimal crucial word over an n-letter alphabet An = {1, 2, . . . , n} avoiding abelian squares has length 4n - 7 for n 3. In this paper, we show 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...

• Type: Journal
• Year: 2009
• Where: DLT
• Authors: Amy Glen, Bjarni V. Halldórsson, Sergey Kitaev
• Keywords: Abelian K-th Powers, Crucial Words, DLT 2009, Minimal Crucial Word, Theoretical Computer Science