Universal Lyndon Words
Arturo Carpi, Gabriele Fici, Stepan Holub, Jakub Oprsal, Marinella, Sciortino

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 over an alphabet is a Lyndon word if there exists an order defined on for which is lexicographically smaller than all of its conjugates (other than itself). We introduce and study \emph{universal Lyndon words}, which are words over an -letter alphabet that have length and such that all the conjugates are Lyndon words. We show that universal Lyndon words exist for every 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.
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Taxonomy
TopicsCoding theory and cryptography · semigroups and automata theory · Algorithms and Data Compression
