A Graph Theory of Rook Placements

Two boards are rook equivalent if they have the same number of non-attacking rook placements for any number of rooks. Define a rook equivalence graph on an equivalence class of Ferrers boards by specifying that two boards are connected by an edge if you can obtain one of the boards by moving squares in the other board out of one column and into a single other column. Given such a graph, we characterize which boards will yield connected graphs. We also provide some cases where common graphs will or will not be the graph for some set of rook equivalent Ferrers boards. Finally, we extend this graph definition to the m-level rook placement generalization developed by Briggs and Remmel. This yields a graph on the set of rook equivalent singleton boards, and we characterize which singleton boards give rise to a connected graph.

Classification of Ding's Schubert Varieties: Finer Rook Equivalence

K. Ding studied a class of Schubert varieties Xƛ in type A partial flag manifolds, indexed by integer partitions ƛ and in bijection with dominant permutations. He observed that the Schubert cell structure of Xƛ is indexed by maximal rook placements on the Ferrers board Bƛ, and that the integral cohomology groups H*(Xƛ; ℤ), H*(Xμ; ℤ) are additively isomorphic exactly when the Ferrers boards Bƛ, Bμ satisfy the combinatorial condition of rook-equivalence.We classify the varieties Xƛ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.