Symmetric products of surfaces and the cycle index

We express the signature $\mathrm{S}\mathrm{i}\mathrm{g}\mathrm{n}(S{P}_G^m(M))$ of the symmetric product $S{P}^n(M)$ of an (open) surface $M$ in terms of the cycle index $Z(G;\bar{x})$ of $G$, a polynomial which originally appeared in P{' o}lya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfaces $M_{g,k},M_{g^{\mathrm{\prime }},k^{\mathrm{\prime }}}$ such that the manifolds $S{P}^m(M_{g,k})$ and $S{P}^m(M_{g^{\mathrm{\prime }},k^{\mathrm{\prime }}})$ are often not homeomorphic, although they always have the same homotopy type provided $2g+k=2g^{\mathrm{\prime }}+k^{\mathrm{\prime }}$ and $k,k^{\mathrm{\prime }}\ge 1$.