Necklaces and Lyndon words in colexicographic order

Abstract

We present the first constant-amortized-time algorithms for generating all length-$n$ necklaces and Lyndon words over a $k$-letter alphabet in colexicographic order, for arbitrary $k\ge 2$. Our approach introduces a novel class of words called \emph{quasinecklaces}, which serve as an easily generated superset of necklaces through which all necklaces can be efficiently identified. We derive a formula for the number $Q_k(n)$ of length-$n$ quasinecklaces and show that $Q_k(n)$ is proportional to the number of length-$n$ necklaces, which is the key property needed to achieve constant amortized time. We also apply our results to efficiently generate a well-known de Bruijn sequence and efficiently generate necklaces and Lyndon words subject to a weight constraint.