Abelian Anti-Powers in Infinite Words

TL;DR

This paper investigates the presence of abelian anti-powers in infinite words, demonstrating that paperfolding words contain abelian anti-powers of all orders, extending previous results on abelian powers.

Contribution

It proves that paperfolding words contain abelian anti-powers of every order, generalizing prior work on abelian powers in such words.

Findings

Paperfolding words contain abelian anti-powers of all orders.

Extension of previous results on abelian powers to anti-powers.

Demonstrates the ubiquity of abelian anti-powers in certain infinite words.

Abstract

An abelian anti-power of order $k$ (or simply an abelian $k$-anti-power) is a concatenation of $k$ consecutive words of the same length having pairwise distinct Parikh vectors. This definition generalizes to the abelian setting the notion of a $k$-anti-power, as introduced in [G. Fici et al., Anti-powers in infinite words, J. Comb. Theory, Ser. A, 2018], that is a concatenation of $k$ pairwise distinct words of the same length. We aim to study whether a word contains abelian $k$-anti-powers for arbitrarily large $k$. S. Holub proved that all paperfolding words contain abelian powers of every order [Abelian powers in paper-folding words. J. Comb. Theory, Ser. A, 2013]. We show that they also contain abelian anti-powers of every order.