On Avoiding Sufficiently Long Abelian Squares

TL;DR

This paper characterizes abelian squares in binary sequences using lattice paths and constructs long binary words that avoid large abelian squares, establishing the asymptotic maximum length of such words.

Contribution

It introduces a lattice path approach to analyze abelian squares and constructs long binary words avoiding large abelian squares, improving understanding of their maximal length.

Findings

Constructed binary words of length q(q+1) avoiding large abelian squares

Proved the maximum length of binary words avoiding abelian squares of length 2k is Θ(k^2)

Characterized abelian squares in binary sequences via Cartesian lattice paths

Abstract

A finite word $w$ is an abelian square if $w = xx^\prime$ with $x^\prime$ a permutation of $x$. In 1972, Entringer, Jackson, and Schatz proved that every binary word of length $k^2 + 6k$ contains an abelian square of length $\geq 2k$. We use Cartesian lattice paths to characterize abelian squares in binary sequences, and construct a binary word of length $q(q+1)$ avoiding abelian squares of length $\geq 2\sqrt{2q(q+1)}$ or greater. We thus prove that the length of the longest binary word avoiding abelian squares of length $2k$ is $\Theta(k^2)$.