  In order to understand the parameter space of monic and centered complex
polynomial vector fields of degree d in the complex plane, decomposed by the
combinatorial classes of the vector fields, it is interesting to know the
number of loci in parameter space consisting of vector fields with the same
combinatorial data (corresponding to topological classification with fixed
separatrices at infinity).
  This paper answers questions posed by Adam L. Epstein and Tan Lei about the
total number of combinatorial classes and the number of combinatorial classes
corresponding to loci of a specific (real) dimension q in parameter space, for
fixed degree d. These results are extensions of a result by Douady, Estrada,
and Sentenac, which shows that the number of combinatorial classes of the
structurally stable complex polynomial vector fields of degree d in the complex
plane is the Catalan number C(d-1). We show that enumerating the combinatorial
classes is equivalent to a so-called bracketing problem. Then we analyze the
generating functions and find closed-form expressions for the number of
classes, as functions of d and q, and we furthermore make an asymptotic
analysis of these sequences for d tending to infinity.
  These results are also applicable to special classes of Abelian
differentials, quadratic differentials with double poles, and singular
holomorphic foliations of the plane.
