Avoiding 2-binomial squares and cubes

TL;DR

This paper investigates the avoidance of 2-binomial squares and cubes in words generated by morphic rules, establishing minimal alphabet sizes for their avoidance, which are proven to be optimal.

Contribution

It proves that 2-binomial squares and cubes are avoidable over minimal alphabets in the context of pure morphic words, extending combinatorial word theory.

Findings

2-binomial squares are avoidable over a 3-letter alphabet

2-binomial cubes are avoidable over a 2-letter alphabet

The alphabet sizes used are proven to be optimal

Abstract

Two finite words $u,v$ are 2-binomially equivalent if, for all words $x$ of length at most 2, the number of occurrences of $x$ as a (scattered) subword of $u$ is equal to the number of occurrences of $x$ in $v$. This notion is a refinement of the usual abelian equivalence. A 2-binomial square is a word $uv$ where $u$ and $v$ are 2-binomially equivalent. In this paper, considering pure morphic words, we prove that 2-binomial squares (resp. cubes) are avoidable over a 3-letter (resp. 2-letter) alphabet. The sizes of the alphabets are optimal.