  Recently, Debruyne and Tenenbaum proved asymptotic formulas for the number of
partitions with parts in $\mathcal{L}\subset\mathbb{N}$ ($\gcd(\mathcal{L})=1$)
and good analytic properties of the corresponding zeta function, generalizing
work of Meinardus. In this paper, we extend their work to prove asymptotic
formulas if $\mathcal{L}$ is a multiset of integers and the zeta function has
multiple poles. In particular, our results imply an asymptotic formula for the
number of irreducible representations of degree $n$ of $\mathfrak{so}{(5)}$. We
also study the Witten zeta function $\zeta_{\mathfrak{so}{(5)}}$, which is of
independent interest.
