  We consider a Markov chain on the space of (countable) partitions of the
interval [0,1], obtained first by size biased sampling twice (allowing
repetitions) and then merging the parts (if the sampled parts are distinct) or
splitting the part uniformly (if the same part was sampled twice). We prove a
conjecture of Vershik stating that the Poisson-Dirichlet law with parameter
theta=1 is the unique invariant distribution for this Markov chain.
  Our proof uses a combination of probabilistic, combinatoric, and
representation-theoretic arguments.
