Words with the Maximum Number of Abelian Squares

TL;DR

This paper investigates infinite words where the count of abelian square factors of length n grows quadratically, exploring the combinatorial properties and maximum occurrences of such structures.

Contribution

It introduces new insights into the growth rate of abelian square factors in infinite words, highlighting conditions for quadratic growth.

Findings

Number of abelian square factors can grow quadratically with word length

Infinite words can be constructed with maximal abelian square occurrences

Provides bounds and characterizations for abelian square distributions

Abstract

An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain $\Theta(n^2)$ distinct factors that are abelian squares. We study infinite words such that the number of abelian square factors of length $n$ grows quadratically with $n$.