New Results on Nyldon Words and Nyldon-like Sets

Grinberg defined Nyldon words as those words which cannot be factorized into a sequence of lexicographically nondecreasing smaller Nyldon words. He was inspired by Lyndon words, defined the same way except with "nondecreasing" replaced by "nonincreasing." Charlier, Philibert, and Stipulanti proved that, like Lyndon words, any word has a unique nondecreasing factorization into Nyldon words. They also show that the Nyldon words form a right Lazard set, and equivalently, a right Hall set. In this paper, we provide a new proof of unique factorization into Nyldon words related to Hall set theory and resolve several questions of Charlier et al. In particular, we prove that Nyldon words of a fixed length form a circular code, we prove a result on factorizing powers of words into Nyldon words, and we investigate the Lazard procedure for generating Nyldon words. We show that these results generalize to a new class of Hall sets, of which Nyldon words are an example, that we name "Nyldon-like sets."Comment: 22 pages; generalized many results to Nyldon-like set

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

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. If the lexicographic order is reversed in this definition, we obtain a new family of words, which are called the Nyldon words. In this paper, 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. Then we investigate this new family of words. In particular, we show that Nyldon words form a right Lazard set

Nyldon words

The theorem of Chen-Fox-Lyndon states that every finite word can be uniquely factorized as a nonincreasing sequence of Lyndon words with respect to the lexicographic order. This theorem can be used to define the family of Lyndon words in a recursive way: 1) the letters are Lyndon; 2) a finite word of length greater than one is Lyndon if it cannot be factorized into a nonincreasing sequence of shorter Lyndon words. In a post on Mathoverflow in November 2014, Darij Grinberg defines a variant of Lyndon words, which he calls Nyldon words, by reversing the lexicographic order in the previous recursive definition. The class of words so obtained is not, as one might first think, the class of maximal words in their conjugacy classes. Gringberg asks three questions: 1) How many Nyldon words of length n are there? 2) Is there an equivalent to the Chen-Fox-Lyndon theorem for Nyldon words? 3) Is it true that every primitive words admits exactly one Nyldon word in his conjugacy class? In this talk, I will discuss these questions in the more general context of Lazard factorizations of the free monoid and show that each of Grinberg’s questions has an explicit answer. This is a joint work with Manon Philibert (ENS Lyon) and Manon Stipulanti (ULiège

Nyldon words

The theorem of Chen-Fox-Lyndon states that every finite word can be uniquely factorized as a nonincreasing sequence of Lyndon words with respect to the lexicographic order. This theorem can be used to define the family of Lyndon words in a recursive way: 1) the letters are Lyndon; 2) a finite word of length greater than one is Lyndon if it cannot be factorized into a nonincreasing sequence of shorter Lyndon words. In a post on Mathoverflow in November 2014, Darij Grinberg defines a variant of Lyndon words, which he calls Nyldon words, by reversing the lexicographic order in the previous recursive definition. The class of words so obtained is not, as one might first think, the class of maximal words in their conjugacy classes. Gringberg asks three questions: 1) How many Nyldon words of length n are there? 2) Is there an equivalent to the Chen-Fox-Lyndon theorem for Nyldon words? 3) Is it true that every primitive words admits exactly one Nyldon word in his conjugacy class? In this talk, I will discuss these questions in the more general context of Lazard factorizations of the free monoid and show that each of Grinberg’s questions has a positive answer. This is a joint work with Manon Philibert (ENS Lyon) and Manon Stipulanti (ULiège)