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27 Graves Place, Holland, MI 49423-3617
In combinatorics, the theory of rooks deals with placements of non-attacking rooks on generalized chess boards and counting such placements. Even though the theory was named after the chess piece rook attacking along rows and columns, the theory was in fact developed to count permutations with restrictions. In this talk, we will investigate the rook theory in three and higher dimensions, inspired by the question “What happens if the rooks fly?” We will show that this theory leads to two different generalizations of the rook placement modeling of the Stirling numbers of the second kind on the triangular boards, starting with two answers to the question “What is a triangular board in three dimensions?” The two answers are intricately related leading to nice results about generalized central factorial and generalized Genocchi numbers. The talk will assume no background in chess or rook theory, and will be a pictorial adventure in combinatorics.
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