Avoiding Letter Patterns in Ternary Square-Free Words

Abstract

We consider special patterns of lengths 5 and 6 in a ternary alphabet. We show that some of them are unavoidable in square-free words and prove avoidability of the other ones. Proving the main results, we use Fibonacci words as codes of ternary words in some natural coding system and show that they can be decoded to square-free words avoiding the required patterns. Furthermore, we estimate the minimal local (critical) exponents of square-free words with such avoidance properties.

Subword complexity and power avoidance

“…An example of such word is the 1-3-bonacci word F 13 obtained similar to the 1-2-bonacci word: take the Fibonacci word f , replace all 0's with 3's to get the codewalk f 13 and take the word with this codewalk as F 13 . The critical exponent of F 13 is 5+ [25]; the fact that F 13 is symmetric and the equality p F 13 (n) = 6n for all n ≥ 5 can be proved as in Theorem 23.…”

“…Now consider the 1-2-bonacci word F 12 ∈ Σ ω 3 , which is the word beginning with 01 and having the codewalk f 12 . This word was introduced by Petrova [25], who proved that F 12 has critical exponent 11/6 (reachable) and no length-5 factors of the form abcab. Also, F 12 appeared to have a nice extremal property [15,Proposition 13].…”

A Ternary Square-free Sequence Avoiding Factors Equivalent to $abcacba$

“…• Let w be the lexicographically least square-free ω-word over {a, b, c}. As the author [1] has pointed out, the method of Shelton [7] allows one to test whether a given finite word over {a, b, c} is a prefix of w. Interest in words avoiding patterns continues, and a recent paper by Petrova [6] studied letter pattern avoidance by ternary square-free words. A word w over {1, 2, 3} avoids the letter pattern P ∈ {x, y, z} * if no factor of w is an image of P under an injection from {x, y, z} to {1, 2, 3}.…”