A ternary square-free sequence avoiding factors equivalent to $abcacba$

TL;DR

This paper classifies and constructs ternary square-free words avoiding a specific letter pattern, demonstrating their existence, characterizing their structure, and showing exponential growth in their count.

Contribution

It completes the classification of avoidable patterns in ternary square-free words and characterizes all such infinite words avoiding a complex pattern.

Findings

Existence of ternary square-free words avoiding the pattern

Characterization of all such infinite words

Exponential growth in the number of these words with length

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, we: (1) characterize all the (two-way) infinite ternary square-free words avoiding letter pattern $xyzxzyx$ (2) characterize the lexicographically least (one-way) infinite ternary square-free word avoiding letter pattern $xyzxzyx$ (3) show that the number of ternary square-free words of length $n$ avoiding letter pattern $xyzxzyx$ grows exponentially with $n$.