Enumerating intermediate-size Boolean antichains in Tamari lattices and generalizing the construction

Determine a recursive formula for the number of Boolean antichains of size k in the Tamari lattice T_{n+1} for 2 ≤ k < n, generalize the construction of Boolean antichains to broader classes of lattices such as geometric, distributive, or modular lattices, and generalize the use of lattice congruences to other settings.

Background

The paper completely enumerates Boolean antichains of every size in Boolean lattices and certain maximal-size Boolean antichains in partition and Tamari lattices. For intermediate sizes in the Tamari lattice, the authors establish structural characterizations through the notions of almost Boolean antichains, closing index sets, and lattice congruences, but do not derive a counting formula.

The authors also identify broader structural questions: whether the recursive methods extend beyond Tamari lattices, particularly to geometric, distributive, and modular lattices, and whether the lattice-congruence framework developed for the Tamari lattice can be applied in other settings. These questions concern both enumeration and the scope of the underlying construction method.