The work enumerates cokernel-closed, idempotent split, full additive subcategories of finite-dimensional complex representations of a uniformly oriented A_n quiver. This integer sequence was identified by Krause and Stoye, and only its first terms were known.

Indecomposable representations correspond to intervals, so a subcategory becomes a subset of a triangular array of points. Cokernel closure translates into a combinatorial closure condition on these subsets. Threshold vectors are introduced and linked to a classical Catalan-number model. This yields a recurrence over threshold vectors, and the underlying lattice of closed subsets is analyzed.

Implication states at level m are counted by the Catalan number C_{m+2}. An implicit recurrence gives terms of the sequence up to n=14. The lattice is atomic and atomistic with n+1 coatoms, and its meet-irreducible elements are described explicitly. The results of the original version were formalized in Lean 4 by a third party.