OFFSET
1,3
COMMENTS
For n>=4, a(n) >= 2n^2 - 8n + 10 which is satisfied by any permutation with the following four points: {(1,2), (2,n), (n-1,1), (n,n-1)}. - Peter Kagey, Oct 30 2021
FORMULA
Conjecture: a(n) = 2n^2 - 8n + 10 for n >= 4. - Peter Kagey, Oct 30 2021
EXAMPLE
For n=3, the two permutations (sets of points) {(1,1),(2,2),(3,3)} and {(1,3),(2,2),(3,1)} have a convex hull with zero area, whereas the remaining four permutations {(1,1),(2,3),(3,2)}, {(1,2),(2,1),(3,3)}, {(1,2),(2,3),(3,1)}, and {(1,3),(2,1),(3,2)} each have a convex hull with area 3/2. So a(3)=3.
MATHEMATICA
(* v. 8.0 *)<<ComputationalGeometry`; a={}; For[n=1, n<=8, n++, {Print[n]; p=Permutations[Range[n]]; an={}; For[k=1, k<=Length[p], k++, {pk=p[[k]]; spk = Table[{i, pk[[i]]}, {i, 1, n}]; AppendTo[an, ConvexHullArea[spk]] }]; AppendTo[a, Max[an]] }]; Print[2*a];
CROSSREFS
KEYWORD
nonn,more
AUTHOR
John W. Layman, Mar 03 2012
STATUS
approved