Closed Ziv-Lempel factorization of the $m$-bonacci words

TL;DR

This paper introduces the concept of closed Ziv-Lempel factorization for words and determines this factorization for infinite m-bonacci words, also classifying their closed prefixes, thus extending factorization theory.

Contribution

It defines the closed Ziv-Lempel factorization and explicitly finds it for all infinite m-bonacci words, a novel extension of factorization methods.

Findings

Closed Ziv-Lempel factorization of m-bonacci words is explicitly characterized.

Classification of closed prefixes of infinite m-bonacci words is provided.

The concept extends factorization techniques to a broader class of words.

Abstract

A word $w$ is said to be closed if it has a proper factor $x$ which occurs exactly twice in $w$, as a prefix and as a suffix of $w$. Based on the concept of Ziv-Lempel factorization, we define the closed $z$-factorization of finite and infinite words. Then we find the closed $z$-factorization of the infinite $m$-bonacci words for all $m \geq 2$. We also classify closed prefixes of the infinite $m$-bonacci words.