Universal Lyndon Words

TL;DR

This paper introduces universal Lyndon words, explores their properties, and provides an algorithm for constructing all such words, revealing their combinatorial structure and connection to Hamiltonian lex-codes.

Contribution

It defines universal Lyndon words, proves their existence for all n, and establishes a bijection with Hamiltonian lex-codes, along with an algorithm for their construction.

Findings

Universal Lyndon words exist for every n.

A bijection between Hamiltonian lex-codes and universal Lyndon words is established.

An algorithm for constructing all universal Lyndon words is provided.

Abstract

A word $w$ over an alphabet $\Sigma$ is a Lyndon word if there exists an order defined on $\Sigma$ for which $w$ is lexicographically smaller than all of its conjugates (other than itself). We introduce and study \emph{universal Lyndon words}, which are words over an $n$-letter alphabet that have length $n!$ and such that all the conjugates are Lyndon words. We show that universal Lyndon words exist for every $n$ and exhibit combinatorial and structural properties of these words. We then define particular prefix codes, which we call Hamiltonian lex-codes, and show that every Hamiltonian lex-code is in bijection with the set of the shortest unrepeated prefixes of the conjugates of a universal Lyndon word. This allows us to give an algorithm for constructing all the universal Lyndon words.