A random walk on the rook placements on a Ferrer's board

A random walk on the rook placements on a Ferrers board

Let B be a Ferrers board, i.e., the board obtained by removing the Ferrers diagram of a partition from the top right corner of an n × n chessboard. We consider a Markov chain on the set R of rook placements on B in which you can move from one placement to any other legal placement obtained by switching the columns in which two rooks sit. We give sharp estimates for the rate of convergence of th...

Invisible permutations and rook placements on a Ferrers board

Derangements on a Ferrers board

We study the derangement number on a Ferrers board B = (n × n) − λ with respect to an initial permutation M , that is, the number of permutations on B that share no common points with M . We prove that the derangement number is independent of M if and only if λ is of rectangular shape. We characterize the initial permutations that give the minimal and maximal derangement numbers for a general F...

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...

Bijections on m-level rook placements

Suppose the rows of a board are partitioned into sets of m rows called levels. An m-level rook placement is a subset of the board where no two squares are in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of m-level rook placements. The first generalizes...