  We consider shifts $\Pi_{n,m}$ of a partially exchangeable random partition
$\Pi_\infty$ of $\mathbb{N}$ obtained by restricting $\Pi_\infty$ to
$\{n+1,n+2,\dots, n+m\}$ and then subtracting $n$ from each element to get a
partition of $[m]:= \{1, \ldots, m \}$. We show that for each fixed $m$ the
distribution of $\Pi_{n,m}$ converges to the distribution of the restriction to
$[m]$ of the exchangeable random partition of $\mathbb{N}$ with the same ranked
frequencies as $\Pi_\infty$. As a consequence, the partially exchangeable
random partition $\Pi_\infty$ is exchangeable if and only if $\Pi_\infty$ is
stationary in the sense that for each fixed $m$ the distribution of $\Pi_{n,m}$
on partitions of $[m]$ is the same for all $n$. We also describe the evolution
of the frequencies of a partially exchangeable random partition under the shift
transformation. For an exchangeable random partition with proper frequencies,
the time reversal of this evolution is the heaps process studied by Donnelly
and others.

Probability