Lyndon pairs and the lexicographically greatest perfect necklace

TL;DR

This paper introduces Lyndon pairs to construct the lexicographically greatest perfect necklaces, generalizing de Bruijn sequences, and provides a method for creating these necklaces for specific divisibility conditions.

Contribution

It defines Lyndon pairs and uses them to construct the lexicographically greatest $(n,k)$-perfect necklaces for cases where $n$ divides $k$ or vice versa, extending previous de Bruijn sequence methods.

Findings

Constructed the lexicographically greatest $(n,k)$-perfect necklaces for specific divisibility cases.

Generalized the de Bruijn sequence construction using Lyndon words and pairs.

Provided a new combinatorial approach to perfect necklaces based on Lyndon structures.

Abstract

Fix a finite alphabet. A necklace is a circular word. For positive integers $n$ and~$k$, a necklace is $(n,k)$-perfect if all words of length $n$ occur $k$ times but at positions with different congruence modulo $k$, for any convention of the starting position. We define the notion of a Lyndon pair and we use it to construct the lexicographically greatest $(n,k)$-perfect necklace, for any $n$ and $k$ such that $n$ divides~$k$ or $k$ divides~$n$. Our construction generalizes Fredricksen and Maiorana's construction of the lexicographically greatest de Bruijn sequence of order $n$, based on the concatenation of the Lyndon words whose length divide $n$.