On Avoiding Sufficiently Long Abelian Squares

Details On Avoiding Sufficiently Long Abelian Squares On Avoiding Sufficiently Long Abelian Squares

2010-12-02

arxiv.org/abs/1012.0524v1

Mathematics

Combinatorics

5 pages

Scientific paper

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 $\Theta(k^2)$.

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