  We provide a structural generalization of a theorem by Kleiman--Piene,
concerning the enumerative geometry of nodal curves in a complete linear system
|L| on a smooth projective surface S. Provided that r, the number of nodes, is
sufficiently small compared to the ampleness of the linear system, we show
that, under certain assumptions, the number of r-nodal curves passing through
points in general position on S is given by a Bell polynomial in universally
defined integers a_i(S,L), which we identify, using classical intersection
theory, as linear, integral polynomials evaluated in four basic Chern numbers.
Furthermore, we provide a decomposition of the a_i as a sum of three terms with
distinct geometric interpretations, and discuss the relationship between these
polynomials and Kazarian's Thom polynomials for multisingularities of maps.
