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%I #22 Nov 05 2021 01:59:54
%S 0,0,3,10,20,34,52,74
%N Twice the maximum of the areas of the convex hulls of permutations {(1,p(1)), (2,p(2)), ..., (n,p(n))} of {1, 2, ..., n}, considered as points in the plane
%C 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
%F Conjecture: a(n) = 2n^2 - 8n + 10 for n >= 4. - _Peter Kagey_, Oct 30 2021
%e 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.
%t (* 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];
%Y Cf. A156831, A208952.
%K nonn,more
%O 1,3
%A _John W. Layman_, Mar 03 2012
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