Enumeration of RC-lattices with at least four reducible elements

Find, up to isomorphism, all RC-lattices containing at least four reducible elements.

Background

The paper enumerates RC-lattices—finite lattices in which all reducible elements are comparable—by nullity and by the number of reducible elements, using the structure of dismantlable lattices and basic blocks. The results cover RC-lattices with nullity k and r reducible elements in the range 2 ≤ r ≤ 2k, and then aggregate these counts to enumerate all RC-lattices on n elements.

The authors explicitly identify the classification of RC-lattices with at least four reducible elements as a remaining problem, extending the enumeration developed in the paper beyond the cases handled by its stated theorems.