Non-Uniqueness of Minimal Superpermutations

TL;DR

This paper investigates the minimal superpermutations problem, providing constructions that challenge the conjecture of uniqueness and minimal length, revealing a vast number of distinct superpermutations for n >= 5.

Contribution

It demonstrates that if the minimal length conjecture holds, then the superpermutation is not unique, and constructs exponentially many such superpermutations.

Findings

Multiple non-unique minimal superpermutations for n >= 5

Existence of more than doubly-exponentially many distinct superpermutations

Challenges the conjecture of uniqueness and minimal length

Abstract

We examine the open problem of finding the shortest string that contains each of the n! permutations of n symbols as contiguous substrings (i.e., the shortest superpermutation on n symbols). It has been conjectured that the shortest superpermutation has length $\sum_{k=1}^n k!$ and that this string is unique up to relabelling of the symbols. We provide a construction of short superpermutations that shows that, if the conjectured minimal length is true, then uniqueness fails for all n >= 5. Furthermore, uniqueness fails spectacularly; we construct more than doubly-exponentially many distinct superpermutations of the conjectured minimal length.