Classifying Nakayama algebras with a braid group action on $\tau$-exceptional sequences

We characterise those basic and connected Nakayama algebras $\Lambda$ for which the mutation of $\tau$-exceptional sequences respects the braid group relations. We show that this is the case if and only if $\Lambda$ is hereditary or all indecomposable projective $\Lambda$-modules have length at least $|\Lambda|$.