  A random composition of $n$ appears when the points of a random closed set
$\widetilde{\mathcal{R}}\subset[0,1]$ are used to separate into blocks $n$
points sampled from the uniform distribution. We study the number of parts
$K_n$ of this composition and other related functionals under the assumption
that $\widetilde{\mathcal{R}}=\phi(S_{\bullet})$, where $(S_t,t\geq0)$ is a
subordinator and $\phi:[0,\infty]\to[0,1]$ is a diffeomorphism. We derive the
asymptotics of $K_n$ when the L\'{e}vy measure of the subordinator is regularly
varying at 0 with positive index. Specializing to the case of exponential
function $\phi(x)=1-e^{-x}$, we establish a connection between the asymptotics
of $K_n$ and the exponential functional of the subordinator.
