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 $p$. The elliptic rook numbers are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and extended to the $q$-case by Garsia and Remmel. We obtain similar results for 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.
@article{10_37236_6121,
author = {Michael J. Schlosser and Meesue Yoo},
title = {Elliptic rook and file numbers},
journal = {The electronic journal of combinatorics},
year = {2017},
volume = {24},
number = {1},
doi = {10.37236/6121},
zbl = {1355.05047},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6121/}
}
TY - JOUR
AU - Michael J. Schlosser
AU - Meesue Yoo
TI - Elliptic rook and file numbers
JO - The electronic journal of combinatorics
PY - 2017
VL - 24
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/6121/
DO - 10.37236/6121
ID - 10_37236_6121
ER -
%0 Journal Article
%A Michael J. Schlosser
%A Meesue Yoo
%T Elliptic rook and file numbers
%J The electronic journal of combinatorics
%D 2017
%V 24
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/6121/
%R 10.37236/6121
%F 10_37236_6121
Michael J. Schlosser; Meesue Yoo. Elliptic rook and file numbers. The electronic journal of combinatorics, Tome 24 (2017) no. 1. doi: 10.37236/6121
