Fixed-Density Necklaces and Lyndon Words in Cool-lex Order

This paper creates a simple Gray code for fixed-density binary necklaces and Lyndon words. The Gray code is simultaneously a left-shift Gray code and a 2-transposition Gray code and is cyclic with these properties. It also has the advantage of using canonical representations (lexicographically minimum rotations) yielding the first known Gray code for Lyndon words ordered by density. This work solidifies the framework that produces loopless Gray code algorithms for generating combinations [15, 16], balanced parentheses and binary trees [17], and multiset permutations [23] in cool-lex order. A primary application of our new fixeddensity Gray code is in the first explicit construction of fixed-weight de Bruijn cycles [24]. In particular, this paper presents a simple O(n)-time implementation that allows each bit of the de Bruijn cycles to be generated in amortized O(1)-time using O(n) memory.