An elliptic extension of the general product formula for augmented rook boards

TL;DR

This paper extends rook theory by developing an elliptic analogue of a known product formula involving augmented rook boards, unifying and generalizing previous models with new elliptic extensions.

Contribution

It introduces an elliptic extension of the $q$-analogue of Miceli and Remmel's rook theory model, broadening the scope of rook theory with a novel mathematical framework.

Findings

Established an elliptic extension of the $q$-analogue of rook theory.

Derived special cases that recover known rook theory models.

Unified various rook theory product formulas within an elliptic framework.

Abstract

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.