Characterization of the lengths of binary circular words containing no squares other than 00, 11, and 0101

TL;DR

This paper precisely characterizes the lengths of binary circular words that avoid all squares except for 00, 11, and 0101, advancing understanding of non-repetitive sequences in combinatorics on words.

Contribution

It provides an exact characterization of possible lengths for such binary circular words, a novel result in the study of square-free sequences.

Findings

Identifies all possible lengths of binary circular words with restricted square patterns.

Establishes a complete classification of these words based on their lengths.

Contributes to the theory of non-repetitive sequences and combinatorics on words.

Abstract

We characterize exactly the lengths of binary circular words containing no squares other than 00, 11, and 0101. Key words: combinatorics on words, circular words, necklaces, square-free words, non-repetitive sequences