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

TL;DR

This paper develops elliptic extensions of rook and matching models, generalizing classical combinatorial formulas and introducing new elliptic analogues of Stirling numbers, Abel polynomials, and matching numbers.

Contribution

It introduces elliptic extensions of the alpha-parameter and rook models, extending product formulas and defining elliptic analogues of key combinatorial numbers.

Findings

Extended product formulas to the elliptic setting.

Derived elliptic analogues of Stirling numbers and Abel polynomials.

Generalized rook theory for matchings using l-lazy graphs.

Abstract

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.