Cycle-counting rook numbers were introduced by Chung and Graham [8]. Cycle-counting q-rook numbers were introduced by Ehrenborg, Haglund, and Readdy [10] and cycle-counting q-hit numbers were introduced by Haglund [14]. Briggs and Remmel [5] introduced the theory of p-rook and p-hit numbers which is a rook theory model where the rook numbers correspond to partial permutations in Cp oSn, the wre...

Rook theory has been investigated by many people since its introduction by Kaplansky and Riordan in 1946. Goldman, Joichi and White in 1975 showed that the sum over k of the product of the (n− k)-th rook numbers multiplied by the k-th falling factorial polynomials factorize into a product. In the sequel, different types of generalizations and analogues of this product formula have been derived ...

Cycle-counting rook numbers were introduced by Chung and Graham [7]. Cycle-counting q-rook numbers were introduced by Ehrenborg, Haglund, and Readdy [9] and cycle-counting q-hit numbers were introduced by Haglund [14]. Briggs and Remmel [4] introduced the theory of p-rook and p-hit numbers which is a rook theory model where the rook numbers correspond to partial permutations in Cp ≀ Sn, the wre...

Rook pivoting is a relatively new pivoting strategy used in Gaussian elimination (GE). It can be as computationally cheap as partial pivoting and as stable as complete pivoting. This paper shows some new attractive features of rook pivoting. We first derive error bounds for the LU factors computed by GE and show rook pivoting usually gives a highly accurate U factor. Then we show accuracy of th...

We consider several generalizations of rook polynomials. In particular we develop analogs of the theory of rook polynomials that are related to general Laguerre and Charlier polynomials in the same way that ordinary rook polynomials are related to simple Laguerre polynomials.

We consider two types of pattern avoidance in the rook monoid, i.e. the set of 0–1 square matrices with at most one nonzero entry in each row and each column. For one-dimensional rook patterns, we completely characterize monoid elements avoiding a single pattern of length at most three and develop an enumeration scheme algorithm to study rook placements avoiding sets of patterns.

We show that, over an arbitrary field, q-rook monoid algebras are iterated inflations of Iwahori-Hecke algebras, and, in particular, are cellular. Furthermore we give an algebra decomposition which shows a q-rook monoid algebra is Morita equivalent to a direct sum of Iwahori-Hecke algebras. We state some of the consequences for the representation theory of q-rook monoid algebras.

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

background : rooks are distributed all over iran and no information is available in the literature on their parasitic infec­tions. methods : one hundred twenty five rooks were examined at post-mortem for parasitic infections. results : two species of cestodes, 5 species of nematodes and 4 species of protozoa were found of which all were new host and distribution record. conclusion : rooks have ...