Rook theory. I. Rook equivalence of Ferrers boards

Rook Poset Equivalence of Ferrers Boards

A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order. We determine when two Ferrers boards have isomorphic rook posets. Equivalently, we give an exact categorization of when two Ding Schubert varieties have iden...

The Inverse Rook Problem on Ferrers Boards

Rook polynomials have been studied extensively since 1946, principally as a method for enumerating restricted permutations. However, they have also been shown to have many fruitful connections with other areas of mathematics, including graph theory, hypergeometric series, and algebraic geometry. It is known that the rook polynomial of any board can be computed recursively. [19, 18] The naturall...

Augmented Rook Boards and General Product Formulas

There are a number of so-called factorization theorems for rook polynomials that have appeared in the literature. For example, Goldman, Joichi and White [6] showed that for any Ferrers board B = F (b1, b2, . . . , bn), n

Invisible permutations and rook placements on a Ferrers board

Rook Theory and Hypergeometric Series

The number of ways of placing k non-attacking rooks on a Ferrers board is expressed as a hypergeometric series, of a type originally studied by Karlsson and Minton. Known transformation identities for series of this type translate into new theorems about rook polynomials.