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...

Let Πn denote the set of all set partitions of {1, 2, . . . , n}. We consider two subsets of Πn, one connected to rook theory and one associated with symmetric functions in noncommuting variables. Let En ⊆ Πn be the subset of all partitions corresponding to an extendable rook (placement) on the upper-triangular board, Tn−1. Given π ∈ Πm and σ ∈ Πn, define their slash product to be π|σ = π∪(σ+m)...

The rook partition algebra RPk(x) is a semisimple algebra for all but a finite number of values of x ∈ C that arises from looking at what commutes with the action of the symmetric group Sn on U ⊗k, where U is the direct sum of the natural representation and the trivial representation of Sn. We give a combinatorial description of this algebra, construct its irreducible representations, and exhib...

Let n be a positive integer and let n = {1, . . . , n}. Let R be the set of all oneto-one maps σ with domain I (σ ) ⊆ n and range J (σ) ⊆ n. If i ∈ I (σ ) let iσ denote the image of i under σ . There is an associative product (σ, τ ) → στ on R defined by composition of maps: i(στ)= (iσ )τ if i ∈ I (σ ) and iσ ∈ I (τ ). Thus the domain I (στ) consists of all i ∈ I (σ ) such that iσ ∈ I (τ ). The...

We study the zeros of two families of polynomials related to rook theory and matchings in graphs. One of these families is based on the cover polynomial of a digraph introduced by Chung and Graham [ChGr]. Another involves a version of the \hit polynomial" of rook theory, but which applies to weighted matchings in (non-bipartite) graphs. For both of these families we prove a result which is anal...

Two boards are rook equivalent if they have the same number of non-attacking placements for any rooks. Define a equivalence graph on an class Ferrers by specifying that two connected edge you can obtain one moving squares in other board out column and into single column. Given such graph, we characterize which will yield graphs. We also provide some cases where common graphs or not be set board...