Nyldon words

The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically nonincreasing sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. In a Mathoverflow post dating from November 2014, Darij Grinberg defines a variant of Lyndon words, which he calls Nyldon words, by reversing the lexicographic order. In a recent collaboration with Emilie Charlier (University of Liège) and Manon Philibert (Aix-Marseille University), we show that every finite word can be uniquely factorized into a lexicographically nondecreasing sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. In our paper, we investigate this new family of words by presenting some of their properties.