Gradedness of the set of rook placements in A n−1

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