Elliptic Rook and File Numbers

  • Michael J. Schlosser
  • Meesue Yoo
Keywords: Rook numbers, File numbers, $q$-Analogues, Elliptic analogues, Combinatorial identities, Stirling numbers, Lah numbers, Trees


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.

Author Biographies

Michael J. Schlosser, University of Vienna
Department of Mathematics
Meesue Yoo, Sungkyunkwan University
AORC Center
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