We provide a structural generalization of a theorem by Kleiman–Piene, concerning the enumerative geometry of nodal curves in a complete linear system 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 which we identify, using classical intersection theory, as linear, integral polynomials evaluated in four basic Chern numbers. Furthermore, we provide a decomposition of the 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.