Elliptic extensions of the alpha-parameter model and the rook model for matchings

We construct elliptic extensions of the alpha-parameter rook model introduced by Goldman and Haglund and of the rook model for matchings of Haglund and Remmel. In particular, we extend the product formulas of these models to the elliptic setting. By specializing the parameter alpha in our elliptic extension of the alpha-parameter model and the shape of the Ferrers board in different ways, we obtain elliptic analogues of the Stirling numbers of the first kind and of the Abel polynomials, and obtain an a,q-analogue of the matching numbers. We further generalize the rook theory model for matchings by introducing l-lazy graphs which correspond to l-shifted boards, where l is a finite vector of positive integers. The corresponding elliptic product formula generalizes Haglund and Remmel's product formula for matchings already in the non-elliptic basic case.