  In these expository notes, we describe some features of the multiplicative
coalescent and its connection with random graphs and minimum spanning trees. We
use Pitman's proof of Cayley's formula, which proceeds via a calculation of the
partition function of the additive coalescent, as motivation and as a
launchpad. We define a random variable which may reasonably be called the
empirical partition function of the multiplicative coalescent, and show that
its typical value is exponentially smaller than its expected value. Our
arguments lead us to an analysis of the susceptibility of the Erd\H{o}s-R\'enyi
random graph process, and thence to a novel proof of Frieze's \zeta(3)-limit
theorem for the weight of a random minimum spanning tree.
