Michael Schlosser

Affiliation: University of Vienna

Email: michael.schlosser@univie.ac.at

Title Of Talk: Elliptic extension of rook theory


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 further provide elliptic extensions of the $j$-attacking model introduced by Remmel and Wachs, and of Haglund and Remmel's rook theory for matchings of graphs. We actually give an extension of the latter which already generalizes the classical, non-elliptic case. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, $r$-restricted versions of all these, and closed form elliptic enumerations of (perfect and maximal) matchings of (complete) graphs. This is joint work with Meesue Yoo.

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