Let $\Lambda$ be a finite-dimensional algebra over the field $K$. We describe how Buan and Marsh’s $\tau$ exception sequences can be used to “brick label” specific pose sets of broad subcategories of finitely generated $\Lambda$ modules. If $\Lambda$ is representation-oriented, we prove that there exists a total order on the set of bricks that make it an EL label. Motivated by the connections between classical anomalous sequences and non-intersecting partitions, we turn our attention to the study of (well-separated) perfect half-patterns. Such a lattice has a bijection between fully co-irreducible and fully co-irreducible elements, known as row motion or simply “$\kappa$ maps”. Generalizing known results for finite half lattices, we show that the $\kappa$ map accurately determines when a set of fully coupled irreducible elements forms a “canonically coupled representation”. The result is that the corresponding “canonically combined complexes” are flag simplicial complexes, as shown for finite semi-distributed lattices and lattices of the torsion class of finite-dimensional algebras. Finally, for lattices of the torsion class of finite-dimensional algebra, we show how Jasso’s $\tau$ tilt reduction can be encoded using the $\kappa$ map. We use this to define the $\kappa^d$-exception sequence for finite half lattices. These are distinct sequences of fully coupled irreducible elements, which have been proven to specialize in the algebraic setting to the $\tau$ exception sequence.
