Abelian-square factors and binary words

TL;DR

This paper confirms a conjecture related to the structure of Abelian-square factors within binary words, advancing understanding in combinatorics on words.

Contribution

It proves a conjecture by Fici and Mignosi about Abelian-square factors in binary words, providing new insights into their combinatorial properties.

Findings

Conjecture by Fici and Mignosi is proven.

Enhanced understanding of Abelian-square factors in binary words.

Contributes to combinatorics on words theory.

Abstract

In this work, we affirm the conjecture proposed by Gabriele Fici and Filippo Mignosi at the 10th Conference on Combinatorics on Words.

Full text

1 Definitions

Definition 1

A word $w$ is called a $factor$ of a word $u$ if there exists words x, y such that $u=xwy$.

Definition 2

An abelian-square is a word of length $2n$ where the first $n$ symbols form an anagram of the last $n$ symbols.

2 Conjecture

Lemma 1

Let $w$ is a word of length $n$, containing $k$ many distinct abelian-square factors, and with the last symbol is in an abelian-square factor. Then a binary word of length $n$ containing at least $k$ many distinct abelian-square factors, and with the last symbol is in an abelian-square factor, exists.

The binary word will be called a binary image of $w$.

Proof

By induction on $n$. For $n<=2$, the claim is clear.

Assume that the claim holds for a word $w$ of length $n$ and $w^{\prime}$ is a binary image of $w$. So, $wx$ with $x$ equals to the last symbol of $w$, has $k$ many distinct abelian-square factors, and has a length $n+1$. And, $w^{\prime}y$ with $y$ equals to the last symbol of $w^{\prime}$ is a binary image of $wx$ ∎

Conjecture 1

[1] Assume that a word with length $n$, and containing $k$ many distinct abelian-square factors, exists. Then a binary word of length $n$ containing at least $k$ many distinct abelian-square factors exists.

Proof

By induction on $n$. For $n<=2$, the claim is clear.

Assume that the claim holds for a word $w$ of length $n$. So, if the last symbol of $w$ is in a factor, then by lemma 1, $wx$ with $x$ equals to the last symbol of $w$ has a binary image of length $n+1$ with at least $k$ distinct abelian-square factors. If the last symbol is not in a factor, then also by lemma 1, $wx$ has a binary image with at least $k+1$ distinct abelian-square factors

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Acknowledgements

The author acknowledges, the financial support of this work from the Tunisian General Direction of Scientific Research (DGRST).