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Suppose the rows of a board are partitioned into sets of m rows calledlevels. An m-level rook placement is a subset of the board where no two squaresare in the same column or the same level. We construct explicit bijections toprove three theorems about such placements. We start with two bijectionsbetween Ferrers boards having the same number of m-level rook placements. Thefirst generalizes a map by Foata and Sch\"utzenberger and our proof applies toany Ferrers board. This bijection also preserves the m-inversion numberstatistic of an m-level rook placement, defined by Briggs and Remmel. Thesecond generalizes work of Loehr and Remmel. This construction only works for aspecial class of Ferrers boards, but it yields a formula for calculating therook numbers of these boards in terms of elementary symmetric functions.Finally we generalize another result of Loehr and Remmel giving a bijectionbetween boards with the same hit numbers. The second and third bijectionsinvolve the Involution Principle of Garsia and Milne.

Bijections on m-level Rook Placements