Universal Cycles for Weak Orders

TL;DR

This paper establishes the existence of universal and s-overlap cycles for weak orders, extending the concept of universal cycles to new combinatorial objects and their variants.

Contribution

It introduces the first universal and s-overlap cycles for weak orders, including those with fixed height and weight, broadening the scope of universal cycle applications.

Findings

Universal cycles for weak orders are proven to exist.

Existence results for s-overlap cycles for weak orders are established.

Applications to cycles for ordered partitions are demonstrated.

Abstract

Universal cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally by Chung, Diaconis, and Graham in 1990. They have been developed by many authors since, for various combinatorial objects such as strings, subsets, permutations, partitions, vector spaces, and designs. One generalization of universal cycles, which require almost complete overlap of consecutive words, is s-overlap cycles, which relax such a constraint. In this paper we study weak orders, which are relations that are transitive and complete. We prove the existence of universal and s-overlap cycles for weak orders, as well as for fixed height and/or weight weak orders, and apply the results to cycles for ordered partitions as well.