  A natural construction due to K. Ding yields Schubert varieties from Ferrers
boards. The poset structure of the Schubert cells in these varieties is equal
to the poset of maximal rook placements on the Ferrers board under the Bruhat
order. We determine when two Ferrers boards have isomorphic rook posets.
Equivalently, we give an exact categorization of when two Ding Schubert
varieties have identical Schubert cell structures. This also produces a
complete classification of isomorphism types of lower intervals of 312-avoiding
permutations in the Bruhat order.
