Elliptic rook and file numbers

TL;DR

This paper introduces elliptic analogues of rook and file numbers for Ferrers and skyline boards, extending classical and q-analogues with additional parameters and providing new factorization theorems and combinatorial applications.

Contribution

It develops elliptic versions of rook and file numbers, generalizing previous q-analogues with two extra parameters and establishing new factorization and combinatorial identities.

Findings

Elliptic rook numbers satisfy an extended factorization theorem.

Elliptic analogues of Stirling, Lah, Abel, and r-restricted numbers are derived.

Applications include new combinatorial identities and models.

Abstract

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's q-rook numbers by two additional independent parameters a and b, and a nome p. These are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and later was extended to the q-case by Garsia and Remmel. We obtain similar results for our elliptic analogues of Garsia and Remmel's q-file numbers for skyline boards. We also provide an elliptic extension of the j-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and r-restricted versions thereof.