Lyndon pairs and the lexicographically greatest perfect necklace
<|reference_start|>Lyndon pairs and the lexicographically greatest perfect necklace: 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 $n$.<|reference_end|>
@article{becher2024lyndon,
    title={Lyndon pairs and the lexicographically greatest perfect necklace},
    author={Ver'onica Becher and Tom'as Tropea},
    journal={arXiv preprint arXiv:2405.17812},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.17812},
    primaryClass={math.CO cs.DM}
}