Non-mathematician here, sorry if terminology is wrong.
- For a circle with N evenly spaced nodes on the perimeter;
- With chords connecting every node to every other node;
- What is the total number of unique intersections of chords?
'Unique' meaning that intersections of 2+ nodes should only be counted once.
I've been able to figure out that the number of line crossings is given by
where p is the pyramid addition function.
But this doesn't account for the 'uniqueness' criterion.
If it happens that no three chords meet at the same point then the number of intersections is $\binom{n}4$ Otherwise it is less. Can you see why? The maximum happens for $n$ prime. That is perhaps not surprising, though that is far from trivial.
The exact formula depends on the congruence class $\bmod 2520=8*9*5*7.$ You can find a link to the proof in the OEIS entry linked above.