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EXAMPLE
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For n=3, draw a line segment from (0,0) to (1,1), from (1,0) to (2,1), and from (2,0) to (0,1). This would correspond to 2 distinct intersections; namely, these are at (5/3,2/3) and (7/3,1/3).
This case corresponds to the permutation where {x_1,x_2,x_3} is {2,3,1}.
For the other 5 permutations, there are at most 2 distinct intersections. Because of this a(3)=2.
This table displays n, a(n), and the lexicographically earliest permutation for the first 13 positive n:
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n a(n) lexicographically earliest permutation
-- ---- ---------------------------------------
1 0 { 1}
2 1 { 2, 1}
3 2 { 2, 3, 1}
4 5 { 3, 4, 2, 1}
5 8 { 3, 5, 4, 2, 1}
6 13 { 5, 6, 4, 3, 1, 2}
7 17 { 4, 7, 6, 3, 5, 2, 1}
8 23 { 6, 7, 8, 5, 3, 4, 1, 2}
9 30 { 7, 9, 4, 8, 6, 3, 5, 2, 1}
10 39 { 9, 10, 7, 8, 4, 3, 6, 5, 2, 1}
11 47 { 9, 8, 11, 10, 7, 6, 5, 3, 4, 1, 2}
12 57 { 9, 12, 11, 8, 7, 10, 4, 3, 6, 5, 2, 1}
13 67 {10, 13, 12, 9, 8, 11, 5, 4, 7, 6, 3, 2, 1}
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