There are level ternary circular square-free words of length $n$ for $n\ne 5,7,9,10,14,17.$

TL;DR

This paper characterizes the lengths for which level ternary circular square-free words exist, providing a complete understanding of their existence based on combinatorial properties of words.

Contribution

It offers a complete characterization of lengths allowing level ternary circular square-free words, advancing the understanding of non-repetitive sequences in combinatorics on words.

Findings

Level ternary circular square-free words exist for all lengths except 5, 7, 9, 10, 14, 17.

Provides a classification based on combinatorial properties.

Enhances understanding of non-repetitive sequences and circular words.

Abstract

A word is level if each letter appears in it the same number of times, plus or minus 1. We give a complete characterization of the lengths for which level ternary circular square-free words exist. Key words: combinatorics on words, circular words, necklaces, square-free words, non-repetitive sequences