On shortening universal words for multi-dimensional permutations

TL;DR

This paper extends the concept of universal words to multi-dimensional permutations, providing new length bounds for their existence using the idea of incomparable elements, generalizing previous results for the 2-dimensional case.

Contribution

It introduces length bounds for universal words in multi-dimensional permutations, generalizing prior work from 2D to higher dimensions using a novel approach.

Findings

Existence of u-words with specific lengths for d-dimensional permutations

Generalization of previous 2D permutation results to higher dimensions

Application of incomparable elements to prove length bounds

Abstract

A universal word (u-word) for $d$-dimensional permutations of length $n$ is a 2-dimensional word with $d-1$ rows, any size $n$ window of which is order-isomorphic to exactly one permutation of length $n$, and all permutations of length $n$ are covered. It is known that u-words (in fact, even u-cycles, a stronger claim) for $d$-dimensional permutations exist. In this paper, we use the idea of incomparable elements to prove that u-words of length $(n!)^{d-1}+n-1-i(n-1)$, for $d\geq 2$ and $$0\leq i\leq \frac{2^{d-1}}{n-1}\left[(1+(n-1)!)^{d-1}-\left(1+\frac{(n-1)!}{2}\right)^{d-1}\right],$$ for $d$-dimensional permutations of length $n$ exist, which generalizes the respective result of Kitaev, Potapov and Vajnovszki for ``usual'' permutations ($d=2$).