Rook theory focuses on placements of non-attacking rooks on boards of various shapes. An important role is played by the rook numbers which count the number of non-attacking placements of a given number of rooks on a board. Ferrers boards,which are boards indexed by integer partitions, are of particular interest. Briggs and Remmel introduced a generalization of rook placements, called m-level rook placements, where a rook is able to attack a subset of the rows.This manuscript presents generalizations of many of the central results regarding rook placements to the case of m-level rook placements. Goldman, Joichi, and White defined the rook polynomial of a board to be the generating function for the rook numbers of that board in the falling factorial basis. By doing so, they were able to give an elegant factorization of the rook polynomial of a Ferrers board in terms of the various column heights. Briggs and Remmel were able to generalize this factorization to the m-level rook polynomial of a subset of Ferrers boards called singleton boards.We give two factorization theorems for the m-level rook polynomial of a Ferrers board. The first is a generalization of the factorization theorem of Briggs and Remmel, working from similar principles. The second relies on a generalization of transposition which we present, called the l-operator. We are also able to use the factorization to describe a unique representative in any m-level equivalence class of Ferrers boards and count the number of singleton boards in the class..When generalizing the factorization from singleton boards to all Ferrers boards, we preserve the definition of the m-level rook polynomial and alter the factorization to apply to all Ferrers boards. We also consider the dual of this problem, applying the factorization of Briggs and Remmel to all Ferrers boards, then trying to determine what is counted by the coefficients of the polynomial in the m-falling factorial basis. It turns out that the coefficients count weighted file placements on a Ferrers board. We also describe a unique representative in each weighted file placement equivalence class of Ferrers boards, as well as count of the number of Ferrers boards in a given weighted file placement equivalence class.Foata and Schü}tzenberger presented explicit bijections between rook placements on any two rook equivalent Ferrers boards as part of their construction of a unique representative in each equivalence class of Ferrers boards. A key tool in their construction was local transposition. We present analogous bijections between m-level rook placements on any two $m$-level rook equivalent Ferrers boards using the local l-operator.The Garsia-Milne Involution Principle was first used in Garsia and Milne's bijective proof of the Rogers-Ramanujan identities. We use it to construct two types of explicit bijections. The first is an explicit bijection between m-level rook placements on any two m-level rook equivalent singleton boards. The second bijection is between the sets counted by the m-level analogue of hit numbers of any two m-level rook equivalent Ferrers boards, providing a bijective proof that $m$-level equivalent Ferrers boards have the same hit numbers.
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