Hahn polynomials and the Burnside process

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.