The Burnside process is a general Markov chain for sampling unlabeled objects. Two examples of the Burnside process give new algorithms for uniformly sampling contingency tables and integer partitions. Both these algorithms have "lumped" versions. These lumped processes are also Markov chains, but they have substantially lower per-step complexity than the original processes. The speed-up is exponential for contingency tables and quadratic for integer partitions. For integer partitions, combining the lumped process with a "deterministic jump" gives another Markov chain called the reflected Burnside process. Empirically, the reflected Burnside process produces uniform integer partitions of size $n=10^8$ in less than 30 steps.
This talk is based on joint work with Persi Diaconis.