Universal Cycles of Restricted Classes of Words

TL;DR

This paper proves the existence of Universal Cycles for various restricted classes of words, extending the concept beyond unrestricted k-letter words to include non-bijections, equitable words, ranked permutations, and passwords.

Contribution

It introduces new existence proofs for Universal Cycles in restricted word classes, broadening the scope of the concept beyond classical cases.

Findings

Universal Cycles exist for non-bijections

Universal Cycles exist for equitable words under certain conditions

Universal Cycles exist for ranked permutations and passwords

Abstract

It is well known that Universal Cycles of $k$-letter words on an $n$-letter alphabet exist for all $k$ and $n$. In this paper, we prove that Universal Cycles exist for restricted classes of words, including: non-bijections, equitable words (under suitable restrictions), ranked permutations, and "passwords".