On Universal Cycles for new Classes of Combinatorial Structures
Antonio Blanca, Anant P. Godbole · SIAM Journal on Discrete Mathematics · 2011
A universal cycle (u-cycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of u-cycles for collections of subsets, restricted multisets, and lattice paths. For subsets, we show that a u-cycle exists for the k-subsets of an n-set if we let k vary in a non zero length interval. We use this result to construct a “covering” of length $(1+o(1))$$n \choose k$ for all subsets of $[n]$ of size exactly k with a specific formula for the $o(1)$ term. We also show that u-cycles exist for all n-length words over some alphabet $\Sigma,$ which contain all characters from $R \subset \Sigma.$ Using this result we provide u-cycles for encodings of Sperner families of size 2 and proper chains of subsets.