Consider the set of all non-intersecting walks of length $n$ on a $d$ dimensional lattice starting at the origin. Now group the members of this set into conformations, where members $x_i$ and $x_j$ belong to the same conformation if they have the same set of nearest neighbors. Let $C(n,d)$ denote the cardinality of the number of unique conformations.

Does anyone have any insight to the bounds of $C(n,d)$ or prior work that deals with this question?

[Please feel free on rephrasing/clarifying the question with more mathematical rigor]