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Rook-by-rook rook theory: Bijective proofs of rook and hit equivalences
Authors:Nicholas A. Loehr  Jeffrey B. Remmel  
Affiliation:aDepartment of Mathematics, Virginia Tech, 460 McBryde Hall, Blacksburg, VA 24061-0123, USA;bDepartment of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA
Abstract:Suppose μ and ν are integer partitions of n, and N>n. It is well known that the Ferrers boards associated to μ and ν are rook-equivalent iff the multisets [μi+i:1less-than-or-equals, slantiless-than-or-equals, slantN] and [νi+i:1less-than-or-equals, slantiless-than-or-equals, slantN] are equal. We use the Garsia–Milne involution principle to produce a bijective proof of this theorem in which non-attacking rook placements for μ are explicitly matched with corresponding placements for ν. One byproduct is a direct combinatorial proof that the matrix of Stirling numbers of the first kind is the inverse of the matrix of Stirling numbers of the second kind. We also prove q-analogues and p,q-analogues of these results. We also use the Garsia–Milne involution principle to show that for any two rook boards B and B, if B and B are bijectively rook-equivalent, then B and B are bijectively hit-equivalent.
Keywords:Rook equivalence   Hit equivalence   Involution principle   q-Analogues
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