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

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 α 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.

Elliptic Rook and File Numbers

“…The elliptic case is more intricate than the q-case and requires a very careful choice of weights. Our factorization theorem for elliptic rook numbers for matchings of l-lazy graphs in [37] generalizes the factorization theorem by Haglund and Remmel already in the ordinary case and the q-case. In the simplest case, our formula can be used to deduce an elliptic extension of the numbers of perfect matchings of the complete graph K 2n .…”

“…By considering rook placements on shifted Ferrers boards subject to a suitable modification of rook cancellation, Haglund and Remmel [23] developed a q-rook theory for matchings of graphs. We provide an elliptic extension of this rook theory in [37]. We actually consider a more general model there related to matchings on certain graphs which we call "l-lazy graphs" with respect to a N-dimensional vector of positive integers l = (l 1 , .…”

Basic Hypergeometric Summations from Rook Theory

“…For an introduction to classical rook theory, see [4]. A lot of material on generalized rook theory which we survey is borrowed from our papers [18,19,20] on elliptic rook theory. As mentioned in the introduction, the (a; q)-case is just a special case of the elliptic case which admits particularly attractive closed formulas.…”

“…The left-hand side of (3.8) is the result of computing the above weight sum columnwise, and the right-hand side can be obtained by computing the weight of the cells in B and and the cells in the extended part separately. For the details, see [20].…”

“…We achieve this by employing a specific one-variable extension of q-rook theory with extra variable a, which we shall refer to as (a; q)-rook theory for short. (The (a; q)-rook numbers we are dealing with in this paper are actually special cases of the elliptic rook numbers which we recently have considered in [18,19,20]. While the elliptic rook numbers satisfy nice identities, they don't factorize in general.…”