Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Augmented Rook Boards and General Product Formulas. | Augmented Rook Boards and General Product Formulas Brian K. Miceli Department of Mathematics Trinity University One Trinity Place San Antonio TX 78212-7200 bmiceli@ Jeffrey Remmel Department of Mathematics University of California San Diego La Jolla CA 92093-0112. USA remmel@ Submitted Aug 18 2007 Accepted Jun 12 2008 Published Jun 20 2008 Mathematics Subject Classification 05A15 05E05 Abstract 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 bl b2 . bn n n n x bi - i - 1 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 kj 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. Keywords rook theory rook placements generating functions Supported in part by NSF grant DMS 0400507 and DMS 0654060 THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 R85 1 Figure 1 A Ferrers board B F 1 2 2 4 c Bn with n 4. 1 Introduction Let N 0 1 2 . denote the set of natural numbers. For any positive integer a we will set a 1 2 . a . We let Bn n X n be the n by n array of squares. We number the rows of Bn from bottom to top and the columns of Bn from left to right with the numbers 1 . n and refer to the square or cell in the i-th row and j-th column of Bn as the i j
