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Abstract for Phil Hanlon, A random walk on the rook
placements on a Ferrer's 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 
$n\times n$ 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 this Markov chain using 
spectral methods.  As part of this analysis we give a
complete combinatorial description of the eigenvalues of the transition matrix 
for this chain.  We show that two extremes cases of this Markov chain correspond 
to random walks on groups which are analyzed in the literature.  
Our estimates for rates of convergence interpolate between those two results.
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