Augmented rook boards and general product formulas (Q1010809)

Summary: There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White showed that for any Ferrers board \(B = F(b_1, b_2, \dots, b_n), \prod_{i=1}^n (x+b_i-(i-1)) = \sum_{k=0}^n r_k(B) (x)\downarrow_{n-k}\) where \(r_k(B)\) is the \(k\)-th rook number of \(B\) and \((x)\downarrow_k = x(x-1) \dots (x-(k-1))\) is the usual falling factorial polynomial. Similar formulas where \(r_k(B)\) is replaced by some appropriate generalization of the \(k\)-th rook number and \((x)\downarrow_k\) is replaced by polynomials like \((x)\uparrow_{k,j} = x(x+j) \dots (x+j(k-1))\) or \((x)\downarrow_{k,j} = x(x-j) \dots (x-j(k-1))\) can be found in the work of \textit{J. Goldman} and \textit{J. Haglund} [J. Comb. Theory, Ser. A 91, No.\,1-2, 509--530 (2000; Zbl 0992.05009)], \textit{J.B. Remmel} and \textit{M.L. Wachs} [Electron. J. Comb. 11, No.\,1, Res. paper R84 (2004; Zbl 1065.05018)], \textit{J. Haglund} and \textit{J.B. Remmel} [Adv. Appl. Math. 27, No.\,2-3, 438--481 (2001; Zbl 1017.05015)], and \textit{K.S. Briggs} and \textit{J.B. Remmel} [J. Comb. Theory, Ser. A 113, No.\,6, 1138--1171 (2006; Zbl 1096.05007)]. 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.