# Unclustered BWTs of any Length over Non-Binary Alphabets

## TL;DR

This paper proves that for any length and alphabet size at least 3, there exist words with BWTs that are completely unclustered, maximizing the number of runs, contrasting with the binary case where such words' existence is open.

## Contribution

It establishes the existence of unclustered BWTs of any length over alphabets of size at least 3, and provides a lower bound on their quantity.

## Key findings

- Existence of unclustered BWTs for all lengths over alphabets of size ≥ 3.
- Maximally unclustered BWTs have exactly n runs.
- Contrast with the binary case where existence is unresolved.

## Abstract

We prove that for every integer $n > 0$ and for every alphabet $\Sigma_k$ of size $k \geq 3$, there exists a necklace of length $n$ whose Burrows-Wheeler Transform (BWT) is completely unclustered, i.e., it consists of exactly $n$ runs with no two consecutive equal symbols. These words represent the worst-case behavior of the BWT for clustering, since the number of BWT runs is maximized. We also establish a lower bound on their number. This contrasts with the binary case, where the existence of infinitely many completely unclustered BWTs is still an open problem, related to Artin's conjecture on primitive roots.