  Rook theory has been investigated by many people since its introduction by
Kaplansky and Riordan in 1946. Goldman, Joichi and White in 1975 showed that
the sum over $k$ of the product of the $(n-k)$-th rook numbers multiplied by
the $k$-th falling factorial polynomials factorize into a product. In the
sequel, different types of generalizations and analogues of this product
formula have been derived by various authors. In 2008, Miceli and Remmel
constructed a rook theory model involving augmented rook boards in which they
showed the validity of a general product formula which can be specialized to
all other product formulas that so far have appeared in the literature on rook
theory. In this work, we construct an elliptic extension of the $q$-analogue of
Miceli and Remmel's result. Special cases yield elliptic extensions of various
known rook theory models.
