  Meinardus proved a general theorem about the asymptotics of the number of
weighted partitions, when the Dirichlet generating function for weights has a
single pole on the positive real axis. Continuing \cite{GSE}, we derive
asymptotics for the numbers of three basic types of decomposable combinatorial
structures (or, equivalently, ideal gas models in statistical mechanics) of
size $n$, when their Dirichlet generating functions have multiple simple poles
on the positive real axis. Examples to which our theorem applies include ones
related to vector partitions and quantum field theory. Our asymptotic formula
for the number of weighted partitions disproves the belief accepted in the
physics literature that the main term in the asymptotics is determined by the
rightmost pole.
