I have N points $[a_{1n}\; a_{2n} \;a_{3n} \;a_{4n}]$ in 4D space constrained by $|a_{1n}|^2+|a_{2n}|^2=|a_{3n}|^2+|a_{4n}|^2=1, \forall n.$ (Note that this means that we have two complex variables of equal magnitude for each $n$.)
Assume a 3-dimensional plane that includes the origin. This plane partitions the N points into two sets ("above" and "below" the plane). How many different partitions are possible?
We may assume that the N points are randomly chosen (subject to the constraint) so we can assume that all of them are different, etc.
The case of no constraints on the N points is treated by Harding'67.