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

James Currie

doi:10.37236/6003

Abstract

We solve a problem of Petrova, finalizing the classification of letter patterns avoidable by ternary square-free words; we show that there is a ternary square-free word avoiding letter pattern $xyzxzyx$. In fact, wecharacterize all the (two-way) infinite ternary square-free words avoiding letter pattern $xyzxzyx$characterize the lexicographically least (one-way) infinite ternary square-free word avoiding letter pattern $xyzxzyx$show that the number of ternary square-free words of length $n$ avoiding letter pattern $xyzxzyx$ grows exponentially with $n$.

Highlights

A theme in combinatorics on words is pattern avoidance
Theorem 1 and Theorem 2 below characterize good Z-words. These turn out to be in 2-to-1 correspondence with square-free Z-words over U
Let π be the morphism on S∗ generated by π(1) = 1, π(2) = 3, π(3) = 2; this morphism π relabels 2’s as 3’s and vice versa

Summary

Introduction

A theme in combinatorics on words is pattern avoidance. A word w encounters word p if f (p) is a factor of w for some non-erasing morphism f. Petrova gives an almost complete classification of the letter patterns over {x, y, z} which can be avoided by ternary square-free words To do this, she uses the notion of ‘codewalks’, developed by Shur [9] as a generalization of the encodings introduced by Pansiot [6]. She uses the notion of ‘codewalks’, developed by Shur [9] as a generalization of the encodings introduced by Pansiot [6] In addition to her classification, Petrova gives upper and lower bounds on the critical exponents of ternary square-free words avoiding letter patterns xyxzx, xyzxy, and xyxzyz. Characterize the lexicographically least (one-way) infinite ternary square-free word avoiding letter pattern xyzxzyx (Theorem 3). Show that the number of ternary square-free words of length n avoiding letter pattern xyzxzyx grows exponentially with n (Theorem 4)

Preliminaries

Results on good words

Proof of Theorem 2

Proof of Theorem 1

Proof of Theorem 3

Proof of Theorem 4