  We prove that the bounded derived category of the lattice of order ideals of
the product of two ordered chains is fractionally Calabi-Yau. We also show that
these lattices are derived equivalent to higher Auslander algebras of type A.
The proofs involve the study of intervals of the poset that have resolutions
described with antichains having rigid properties. These two results combined
corroborate a conjecture by Chapoton linking posets to Fukaya Seidel
Categories.
