# Computing the $k$-binomial complexity of the Thue--Morse word

## TL;DR

This paper investigates the $k$-binomial complexity of the Thue--Morse word, revealing that despite its aperiodicity, the complexity eventually stabilizes to only two values, which is a surprising and novel finding.

## Contribution

It characterizes the $k$-binomial complexity of the Thue--Morse word and introduces a new equivalence relation to analyze its factors, providing new insights into its combinatorial structure.

## Key findings

- The $k$-binomial complexity of Thue--Morse eventually takes only two values.
- A new equivalence relation for analyzing factors of Thue--Morse is introduced.
- General results on subword occurrences in iterates of morphisms are established.

## Abstract

Two words are $k$-binomially equivalent whenever they share the same subwords, i.e., subsequences, of length at most $k$ with the same multiplicities. This is a refinement of both abelian equivalence and the Simon congruence. The $k$-binomial complexity of an infinite word $\mathbf{x}$ maps the integer $n$ to the number of classes in the quotient, by this $k$-binomial equivalence relation, of the set of factors of length $n$ occurring in $\mathbf{x}$. This complexity measure has not been investigated very much. In this paper, we characterize the $k$-binomial complexity of the Thue--Morse word. The result is striking, compared to more familiar complexity functions. Although the Thue--Morse word is aperiodic, its $k$-binomial complexity eventually takes only two values. In this paper, we first obtain general results about the number of occurrences of subwords appearing in iterates of the form $\Psi^\ell(w)$ for an arbitrary morphism $\Psi$. We also thoroughly describe the factors of the Thue--Morse word by introducing a relevant new equivalence relation.