A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system. Degenerations of such orbits induce a natural partial order on the set of rook placements. We study combinatorial structure of the set of rook placements in An− 1 with respect to a slightly different order and prove that this poset is graded.

• 05E16 - Combinatorial aspects of groups and algebras
• 17B08 - Coadjoint orbits; nilpotent varieties
• 17B22 - Root systems