Characterizations of Perfectly Clustering Words

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).