A Graph Theory of Rook Placements

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