Enumerating Combinatorial Classes of the Complex Polynomial Vector Fields in the Complex Plane

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.