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.