Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bijections on m-level Rook Placements   Suppose the rows of a board are partitioned into sets of m rows called
levels. An m-level rook placement is a subset of the board where no two squares
are in the same column or the same level. We construct explicit bijections to
prove three theorems about such placements. We start with two bijections
between Ferrers boards having the same number of m-level rook placements. The
first generalizes a map by Foata and Sch\"utzenberger and our proof applies to
any Ferrers board. This bijection also preserves the m-inversion number
statistic of an m-level rook placement, defined by Briggs and Remmel. The
second generalizes work of Loehr and Remmel. This construction only works for a
special class of Ferrers boards, but it yields a formula for calculating the
rook numbers of these boards in terms of elementary symmetric functions.
Finally we generalize another result of Loehr and Remmel giving a bijection
between boards with the same hit numbers. The second and third bijections
involve the Involution Principle of Garsia and Milne.

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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn