Infinitely many binary completely unclustered BWTs

Determine whether there exist infinitely many binary necklaces over the alphabet {0,1} whose Burrows–Wheeler Transform is completely unclustered, namely, has exactly |u| runs with no two consecutive equal symbols. Equivalently, ascertain whether the set of lengths n for which a binary necklace of length n has a completely unclustered Burrows–Wheeler Transform is infinite.

Background

The paper proves that for any length n and any alphabet size k ≥ 3 there exists a necklace whose Burrows–Wheeler Transform (BWT) is completely unclustered. This sharply contrasts with the binary case, where such a result is not known.

In the binary setting, Mantaci et al. established that a completely unclustered BWT exists for length 2n if and only if 2n+1 is prime and 2 is a primitive root modulo 2n+1, linking the problem to number theory. The authors emphasize that whether there are infinitely many such binary instances remains unresolved and is tied to Artin’s conjecture on primitive roots.