# Superpermutation matrices

## TL;DR

This paper introduces superpermutation matrices, linking their minimal size to universal words for symmetric group quotients, and establishes bounds and asymptotic behavior for their minimal length.

## Contribution

It proposes superpermutation matrices and connects their properties to universal words, providing bounds and asymptotic analysis for minimal lengths.

## Key findings

- The minimal size of superpermutation matrices relates to universal words for symmetric group quotients.
- Bounds on the minimal length of such universal words are established.
- The ratio of minimal length to the size of the alphabet approaches 2 as n increases.

## Abstract

Superpermutations are words over a finite alphabet containing every permutation as a factor. Finding the minimal length of a superpermutation is still an open problem. In this article, we introduce superpermutations matrices. We establish a link between the minimal size of such a matrix and the minimal length of a universal word for the quotient of the symmetric group $S_n$ by an equivalence relation. We will then give non-trivial bounds on the minimal length of such a word and prove that the limit of their ratio when $n$ approaches infinity is 2.