Primitive Words and Lyndon Words in Automatic and Linearly Recurrent Sequences

TL;DR

This paper explores the properties of primitive and Lyndon words within automatic and linearly recurrent sequences, revealing their automaticity, regularity, and boundedness in various contexts.

Contribution

It demonstrates that Lyndon factorizations of k-automatic sequences are k-automatic and that the counts of primitive and Lyndon factors are k-regular, also establishing bounds for Lyndon factors in linearly recurrent sequences.

Findings

Lyndon factorization of a k-automatic sequence is k-automatic.

Counting primitive and Lyndon factors yields k-regular functions.

Number of Lyndon factors in linearly recurrent sequences is bounded.

Abstract

We investigate questions related to the presence of primitive words and Lyndon words in automatic and linearly recurrent sequences. We show that the Lyndon factorization of a k-automatic sequence is itself k-automatic. We also show that the function counting the number of primitive factors (resp., Lyndon factors) of length n in a k-automatic sequence is k-regular. Finally, we show that the number of Lyndon factors of a linearly recurrent sequence is bounded.