Invisible permutations and rook placements on a Ferrers board

Invisible permutations and rook placements on a Ferrers 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...

Pattern Avoiding Permutations & Rook Placements

First, we look at the distribution of permutation statistics in the context of pattern-avoiding permutations. The first part of this chapter deals with a recursively defined bijection of Robertson [6] between 123and 132-avoiding permutations. We introduce the general notion of permutation templates and pivots in order to give a non-recursive pictorial reformulation of Robertson’s bijection. Thi...

Descents of Permutations in a Ferrers Board

The classical Eulerian polynomials are defined by setting An(t) = ∑ σ∈Sn t = n ∑ k=1 An,kt k where An,k is the number of permutations of length n with k − 1 descents. Let An(t, q) = ∑ π∈Sn t 1+des(π)qinv(π) be the inv q-analogue of the classical Eulerian polynomials whose generating function is well known: ∑ n>0 uAn(t, q) [n]q! = 1 1− t ∑ k>1 (1− t)kuk [k]q! . (0.1) In this paper we consider pe...

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...