Enumeration of Combinatorial Classes of Single Variable Complex Polynomial Vector Fields

A vector field in the space of degree d monic, centered single variable complex polynomial vector fields has a combinatorial structure which can be fully described by a combinatorial data set consisting of an equivalence relation and a marked subset on the integers mod 2d-2, satisfying certain properties, corresponding to an equivalence relation on the 2d-2 separatrices. This data set can be equivalently represented in a combinatorial disk model, by labeling the 2d-2 roots of unity on the unit circle by the asymptotic directions of the separatrices and joining the points in the same equivalence class by geodesics in the unit disk with respect to the Poincaré metric. The goal is to utilize this fact in order to convert the problem of counting the combinatorial classes into a so-called bracketing problem: a combinatorial problem involving pairings of parentheses placed in a string of elements in a valid way. We first enumerate all combinatorial classes with respect to degree d, and then we enumerate the combinatorial classes having a specific dimension q in parameter space. In both cases, a recursion equation and implicit expressions for the algebraic generating functions are calculated, and asymptotic growth questions are considered.