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arXiv · 1008.2251

On Universal Cycles for new Classes of Combinatorial Structures

Antonio Blanca·Anant P. Godbole
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Abstract

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, matroids, restricted multisets, chains of subsets, multichains, 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 $Σ,$ which contain all characters from $R \subset Σ.$ Using this result we provide u-cycles for encodings of Sperner families of size 2 and proper chains of subsets.

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Antonio Blanca, Anant P. Godbole. 2010-08-13. On Universal Cycles for new Classes of Combinatorial Structures. https://arxiv.org/abs/1008.2251

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