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Lyndon pairs and the lexicographically greatest perfect necklace

Verónica Becher, Tomás Tropea

TL;DR

The paper addresses constructing the lexicographically greatest $(n,k)$-perfect necklace over an alphabet of size $s$, for all $n,k$ with $n|k$ or $k|n$. It generalizes Fredricksen and Maiorana’s de Bruijn construction by introducing Lyndon pairs and an operator $ heta$ to systematically enumerate maximal rotations and assemble a necklace $X$ of length $s^n k$. The main contributions are (1) the formalization of Lyndon pairs and the $ heta$-based traversal, (2) a complete construction of the lexicographically greatest $(n,k)$-perfect necklace for the two divisibility regimes, including detailed case analyses and proofs of length, perfection, and maximality, and (3) a framework that extends de Bruijn sequence generation to broader perfect necklaces with potential applications in normal number constructions and combinatorial sequence design. The approach yields a linear-time, low-space method for generating these necklaces via concatenation of Lyndon words, mirroring the classical FM78 construction while broadening its scope.

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 $n$.

Lyndon pairs and the lexicographically greatest perfect necklace

TL;DR

The paper addresses constructing the lexicographically greatest -perfect necklace over an alphabet of size , for all with or . It generalizes Fredricksen and Maiorana’s de Bruijn construction by introducing Lyndon pairs and an operator to systematically enumerate maximal rotations and assemble a necklace of length . The main contributions are (1) the formalization of Lyndon pairs and the -based traversal, (2) a complete construction of the lexicographically greatest -perfect necklace for the two divisibility regimes, including detailed case analyses and proofs of length, perfection, and maximality, and (3) a framework that extends de Bruijn sequence generation to broader perfect necklaces with potential applications in normal number constructions and combinatorial sequence design. The approach yields a linear-time, low-space method for generating these necklaces via concatenation of Lyndon words, mirroring the classical FM78 construction while broadening its scope.

Abstract

Fix a finite alphabet. A necklace is a circular word. For positive integers and~, a necklace is -perfect if all words of length occur times but at positions with different congruence modulo , for any convention of the starting position. We define the notion of a Lyndon pair and we use it to construct the lexicographically greatest -perfect necklace, for any and such that divides~ or divides~. Our construction generalizes Fredricksen and Maiorana's construction of the lexicographically greatest de Bruijn sequence of order , based on the concatenation of the Lyndon words whose length divide .
Paper Structure (18 sections, 7 theorems, 75 equations)

This paper contains 18 sections, 7 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Lyndon pairs in lexicographical order
  3. Lyndon pairs
  4. The operator $\theta$
  5. Statement of Theorem \ref{['thm:perfectp']}
  6. Proof of Theorem \ref{['thm:perfectp']}
  7. Proof that necklace $X$ has length $s^n k$
  8. When $k|n, k< n$
  9. When $n|k$, $n\leq k$
  10. Proof that necklace $X$ is $(n,k)$-perfect
  11. When $k|n, k< n$
  12. Case $A = (s-1)^n$.
  13. Case $A = 0^n$.
  14. Case $(s-1)^n > A >0^n$
  15. When $n|k$, $n\leq k$
  16. ...and 3 more sections

Key Result

Lemma 1

Let $n$ and $k$ be positive integers such that $k|n$ or $n|k$. Let ${\EuScript A} = \langle A, 0\rangle$ in $\Sigma^n\times \mathbb{Z}_k$ be maximal and different from $\langle0^n, 0\rangle$. Suppose $A=a_1\ldots a_n$, let $i$ be such that $a_i>0$ and let where if $n|k$ then $j=n$; and if $k|n$ then $j$ is the smallest such that $k|j$ and $i\leq j< n$. Then, ${\EuScript B}\in\Sigma^j\times \mathb

Theorems & Definitions (26)

  • Definition : order $\succ$ on $\Sigma^n\times\mathbb{Z}_k$
  • Definition : rotation of a pair
  • Definition : maximal pair
  • proof
  • Lemma 1
  • proof
  • Definition : reduction
  • Definition : expansion
  • Definition : Lyndon pair
  • Definition : operator $\theta$
  • ...and 16 more