I am thinking of two configurations of rings as being equivalent if you can physically move them from one configuration to the other without deforming the rings or have them intersect each-other.
If you think of each ring as a node on a graph and the linking of two rings as edges of a graph then this problem becomes how many graphs of order n are there. Or maybe I am missing something, like if ring C can go through rings A and B in multiple ways ...
The only issue I can see is if there is some graph for which there does not correspond an allowable ( edges of rings get in the way somehow) configuration of rings. I know this is the difficulty for calculating the number of possible Venmo diagrams (n=6 isn’t even known!) but not sure if the same issue arises with this problem.