  In the study of the rational cohomology of Hilbert schemes of points on a
smooth surface, it is particularly interesting to understand the characteristic
classes of the tautological bundles and the tangent bundle. In this note we
pursue this study. We first collect all results appearing separately in the
literature and prove some new formulas using T. Ohmoto's results on orbifold
Chern classes on Hilbert schemes. We also explain the algorithmic counterpart
of the topic: The cohomology space is governed by a vertex algebra that can be
used to compute characteristic classes. We present an implementation of the
vertex operators in the rewriting logic system {\sc Maude} and address
observations and conjectures obtained after symbolic computations.
