Binary Words Containing Few Abelian Squares

TL;DR

This paper explores the minimal number of abelian squares in binary words, extending a conjecture and providing constructions for words with minimal abelian squares based on Parikh vectors.

Contribution

It extends Fici and Saarela's conjecture and offers specific constructions for binary words with minimal abelian squares based on Parikh vectors.

Findings

The conjecture holds in some special cases.

Constructed words with minimal abelian squares for given Parikh vectors.

Proposed conjecture for minimal abelian squares in binary words.

Abstract

Fici and Saarela ([2]) conjectured that a binary word of length n contains at least $\lfloor n/4 \rfloor$ abelian squares. We slightly extend this conjecture and show that it holds in some special cases. In all other cases we have the following: given a Parikh vector over a two letter alphabet we produce a word with that Parikh vector which we conjecture contains the least possible number of abelian squares.