Hello Pierre Charland,>But I think there is a problem,
>I can't find 22 regions for n=4.
>So the reasonning leading to the formula must be wrong.
The error was mine. In my earlier post, I said that the problem was to find the number of regions for n(n-1)/2 lines in general position. However, what I meant to say was that the lines were in general position, EXCEPT at the original set of n points which determined the lines.
At the original points, there are n-1 concurrent lines, so of course this is not general position. You can account for the change in the number of regions like this:
For each of the original n points, place a small circle around it, small enough that none of the other points is inside the circle. Then displace the lines slightly to remove the concurrence, i.e. put the n-1 lines into general position inside this circle. Now the area inside the circle will be cut into (n-1,0) + (n-1,1) + (n-1,2) regions, according to your formula. However, when the lines were concurrent, they cut this circle into 2(n-1) regions, like slicing a round pie through the center. The difference between these numbers is the number of regions which are removed by the concurrency.
Since (n-1,0) + (n-1,1) + (n-1,2) - 2(n-1) = (n-2)(n-3)/2, the total loss of regions for all n points is n(n-2)(n-3)/2. This formula is true only for n > 1, since for n = 0 or 1, the binomial coefficient formula is not valid. However, for n <= 3 there are at most 2 lines through each point, and so the lines truly are in general position. This is shown also by the fact that the correction vanishes at n = 2 or n = 3.
Therefore, subtracting the correction from the formula you gave for lines in general position, we get the new polynomial
(n^4 - 2n^3 + 3n^2 - 2n + 8)/8 - (n^3 - 5n^2 + 6n)/2 =
(n^4 -6n^3 + 23n^2 +22n +8)/8
This should be correct now. To check, the correction for n = 4 is 4(2)(1)/2 = 4, which reduces the region count from 22 to 18 as it should.
I think this clears up the problem. Sorry for the original unclear formulation.
--Stuart Anderson