Stationary random partitions of positive integers

This paper gives a description of stationary random partitions of positive integers (equivalently, stationary coherent sequences of random permutations) under the action of the infinite symmetric group. Equivalently, all stationary coherent sequences of random permutations are described. This result gives a new characterization of the Poisson–Dirichlet distribution PD(1) with the unit parameter, which turns out to be the unique invariant distribution for a family of Markovian operators on the infinite-dimensional simplex. This result also provides a new characterization of the Haar measure on the projective limit of finite symmetric groups.