On shortening u-cycles and u-words for permutations

TL;DR

This paper explores methods to shorten universal cycles and words for permutations by incorporating incomparable elements or non-deterministic symbols, extending concepts similar to de Bruijn sequences.

Contribution

It introduces new approaches for shortening u-cycles and u-words for permutations using incomparable elements and non-deterministic symbols, providing explicit length constructions.

Findings

Existence of u-words for n-permutations with lengths n!+(1-k)(n-1) for k=0,...,(n-2)!

Extension of shortening techniques to non-deterministic symbols

Connection to recent studies on de Bruijn sequences

Abstract

This paper initiates the study of shortening universal cycles (u-cycles) and universal words (u-words) for permutations either by using incomparable elements, or by using non-deterministic symbols. The latter approach is similar in nature to the recent relevant studies for the de Bruijn sequences. A particular result we obtain in this paper is that u-words for $n$-permutations exist of lengths $n!+(1-k)(n-1)$ for $k=0,1,\ldots,(n-2)!$.