## Augmented rook boards and general product formulas

### Summary

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 [6] showed that for any Ferrers board B = F (b1, b2, . . . , bn), n n (x + bi - (i - 1)) = $rk(B)(x) \downarrow n$ - k i=1 k=0 where $rk(B)$ is the k-th rook number of B and (x) $\downarrow k= x(x - 1) \cdot \cdot \cdot$(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) $\downarrow$is replaced by k polynomials like (x) $\uparrow k,j= x(x + j) \cdot \cdot \cdot (x + j(k - 1))$ or (x) $\downarrow k,j= x(x - j) \cdot \cdot \cdot$(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.

05A15, 05E05

### Keywords/Phrases

rook theory, rook placements, generating functions $\ast$Supported in part by NSF grant DMS 0400507 and DMS 0654060 the electronic journal of combinatorics 15 (2008), #R85