Universal and Overlap Cycles for Posets, Words, and Juggling Patterns

TL;DR

This paper explores the existence of universal and overlap cycles across various combinatorial structures, including posets, words, and juggling patterns, providing new theoretical results in these areas.

Contribution

It proves the existence of universal cycles for naturally labeled posets and s-overlap cycles for words and juggling patterns, including the first known u-cycle with an unknown length.

Findings

Universal cycles for NL posets established

s-overlap cycles for words of fixed weight demonstrated

Existence of u-cycle with unknown length proved

Abstract

We discuss results dealing with universal cycles (u-cycles) and $s$-overlap cycles, and contribute to the body of those results by proving existence of universal cycles of naturally labeled posets (NL posets), $s$-overlap cycles of words of weight $k$, and juggling patterns. The result on posets is, to the best of our knowledge, the first demonstration of the existence of a u-cycle whose length is unknown.