  We study a natural Markov chain on $\{0,1,\cdots,n\}$ with eigenvectors the
Hahn polynomials. This explicit diagonalization makes it possible to get sharp
rates of convergence to stationarity. The process, the Burnside process, is a
special case of the celebrated `Swendsen-Wang' or `data augmentation'
algorithm. The description involves the beta-binomial distribution and Mallows
model on permutations. It introduces a useful generalization of the Burnside
process.
