Generalizing the Classic Greedy and Necklace Constructions of de Bruijn Sequences and Universal Cycles

We present a class of languages that have an interesting property: For each language L in the class, both the classic greedy algorithm and the classic Lyndon word (or necklace) concatenation algorithm provide the lexicographically smallest universal cycle for L. The languages consist of length n strings over {1, 2, . . . , k} that are closed under rotation with their subset of necklaces also being closed under replacing any suffix of length i by i copies of k. Examples include all strings (in which case universal cycles are commonly known as de Bruijn sequences), strings that sum to at least s, strings with at most d cyclic descents for a fixed d > 0, strings with at most d cyclic decrements for a fixed d > 0, and strings avoiding a given period. Our class is also closed under both union and intersection, and our results generalize results of several previous papers.