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Augmented Rook Boards and General Product Formulas

By Brian K. Miceli and Jeffrey Remmel


There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White [6] showed that for any Ferrers board B = F (b1, b2,..., bn), n∏ (x + bi − (i − 1)) = i=1 n∑ rk(B)(x) ↓n−k where rk(B) is the k-th rook number of B and (x) ↓k = x(x − 1) · · · (x − (k − 1)) is the usual falling factorial polynomial. Similar formulas where rk(B) is replaced by some appropriate generalization of the k-th rook number and (x) ↓k is replaced by polynomials like (x) ↑k,j = x(x + j) · · · (x + j(k − 1)) or (x) ↓k,j = x(x − j) · · · (x − j(k − 1)) can be found in the work of Goldman and Haglund [5], Remmel and Wachs [9], Haglund and Remmel [7], and Briggs and Remmel [3]. We shall refer to such formulas as product formulas. The main goal of this paper is to develop a new rook theory setting in which we can give a uniform combinatorial proof of a general product formula that includes, as special cases, essentially all the product formulas referred to above. We shall also prove q-analogues and (p, q)-analogues of our general product formula. k=

Topics: rook theory, rook placements, generating functions
Year: 2010
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