  The plethysm coefficient $p(\nu, \mu, \lambda)$ is the multiplicity of the
Schur function $s_\lambda$ in the plethysm product $s_\nu \circ s_\mu$. In this
paper we use Schur--Weyl duality between wreath products of symmetric groups
and the ramified partition algebra to interpret an arbitrary plethysm
coefficient as the multiplicity of an appropriate composition factor in the
restriction of a module for the ramified partition algebra to the partition
algebra. This result implies new stability phenomenon for plethysm coefficients
when the first parts of $\nu$, $\mu$ and $\lambda$ are all large. In
particular, it gives the first positive formula in the case when $\nu$ and
$\lambda$ are arbitrary and $\mu$ has one part. Corollaries include new
explicit positive formulae and combinatorial interpretations for the plethysm
coefficients $p((n-b,b), (m), (mn-r,r))$, and $p((n-b,1^b), (m), (mn-r,r))$
when $m$ and $n$ are large.
