Shortened universal cycles for permutations

TL;DR

This paper proves a conjecture that universal cycles for permutations can be shortened using incomparable elements, providing a new constructive method for their construction.

Contribution

It confirms a conjecture on shortening universal cycles for permutations and introduces a new constructive approach for their creation.

Findings

Proved that universal cycles for permutations can be shortened to length n!-i(n-1).

Developed a new method for constructing universal cycles for permutations.

Validated the conjecture by Kitaev, Potapov, and Vajnovszki.

Abstract

Kitaev, Potapov, and Vajnovszki [On shortening u-cycles and u-words for permutations, Discrete Appl. Math, 2019] described how to shorten universal words for permutations, to length $n!+n-1-i(n-1)$ for any $i \in [(n-2)!]$, by introducing incomparable elements. They conjectured that it is also possible to use incomparable elements to shorten universal cycles for permutations to length $n!-i(n-1)$ for any $i \in [(n-2)!]$. In this note we prove their conjecture. The proof is constructive, and, on the way, we also show a new method for constructing universal cycles for permutations.