Characterizations of Perfectly Clustering Words

TL;DR

This paper introduces a new factorization method for perfectly clustering words, generalizing known characterizations of Christoffel words and providing insights into their combinatorial structure.

Contribution

It proposes a novel palindrome-based factorization of perfectly clustering words, extending classical characterizations of Christoffel words.

Findings

Factorization into n-1 palindromes with interleaved letters

Generalization of Pirillo's and de Luca-Mignosi's characterizations

Enhanced understanding of the structure of perfectly clustering words

Abstract

Perfectly clustering words are one of many possible generalizations of Christoffel words. In this article, we propose a factorization of a perfectly clustering word on a $n$ letters alphabet into a product of $n-1$ palindromes with a letter between each of them. This factorization allows us to generalize two combinatorial characterization of Christoffel words due to Pirillo (1999) and de Luca and Mignosi (1994).