Contributions to the theory of de Bruijn cycles (1304.2820v1)

Abstract: A de Bruijn cycle is a cyclic listing of length A, of a collection of A combinatorial objects, so that each object appears exactly once as a set of consecutive elements in the cycle. In this paper, we show the power of de Bruijn's original theorem, namely that the cycles bearing his name exist for n-letter words on a k-letter alphabet for all values of k,n, to prove that we can create de Bruijn cycles for the assignment of elements of [n]={1,2,....,n} to the sets in any labeled subposet of the Boolean lattice; de Bruijn's theorem corresponds to the case when the subposet in question consists of a single ground element. The landmark work of Chung, Diaconis, and Graham extended the agenda of finding de Bruijn cycles to possibly the next most natural set of combinatorial objects, namely k-subsets of [n]. In this area, important contributions have been those of Hurlbert and Rudoy. Here we follow the direction of Blanca and Godbole, who proved that, in a suitable encoding, de Bruijn cycles can be created for the subsets of [n$ of size in the interval [s,t]; 0<=s<t<=n$. In this paper we generalize this result to exhibit existence of de Bruijn cycles for words with weight between s and t, where these parameters are suitably restricted.