Prob. Theory Appl.,

**Abstract.**
We describe stationary random partitions of positive integers
(in equivalent setting - of stationary coherent sequences of
random permutations) with respect to a natural action of the
infinite symmetric group. The result yields a new characterization
of the Poisson-Dirichlet distribution PD(1) as the unique
invariant distribution for a family of Markovian operators
on the infinite-dimensional simplex, as well as a new
characterization of the Haar measure on the projective
limit of finite symmetric groups.

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