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arXiv · 1406.5895

Universal Lyndon Words

Arturo Carpi·Gabriele Fici·Stepan Holub·Jakub Oprsal·Marinella Sciortino
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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.

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Arturo Carpi, Gabriele Fici, Stepan Holub, Jakub Oprsal, Marinella Sciortino. 2014-06-23. Universal Lyndon Words. https://arxiv.org/abs/1406.5895

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