Square-Free Shuffles of Words

TL;DR

This paper proves the existence of infinite square-free words over three letters that, when shuffled with themselves, produce other infinite square-free words, advancing understanding of word combinatorics and shuffle operations.

Contribution

It demonstrates the existence of infinite square-free words that can be self-shuffled to generate other square-free words, a novel result in combinatorics on words.

Findings

Existence of square-free words of arbitrary length with self-shuffle properties

Construction of infinite square-free words with self-shuffle to produce infinite square-free words

Extension of finite results to infinite words in the context of square-freeness

Abstract

Let $u \shuffle v$ denote the set of all shuffles of the words $u$ and $v$. It is shown that for each integer $n \geq 3$ there exists a square-free ternary word $u$ of length $n$ such that $u\shuffle u$ contains a square-free word. This property is then shown to also hold for infinite words, i.e., there exists an infinite square-free word $u$ on three letters such that $u$ can be shuffled with itself to produce an infinite square-free word $w \in u \shuffle u$.