  We express the signature ${\rm Sign}(SP^m_G(M))$ of the symmetric product
$SP^n(M)$ 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 $M_{g,k}, M_{g',k'}$ such that the
manifolds $SP^{m}(M_{g,k})$ and $SP^{m}(M_{g',k'})$ are often not homeomorphic,
although they always have the same homotopy type provided $2g+k = 2g'+k'$ and
$k,k'\geq 1$.
