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A208952 Number of distinct values 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. 1

%I

%S 1,1,2,5,11,24,39,60

%N Number of distinct values 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.

%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. Thus there are two distinct values of the areas, so a(3)=2.

%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, Length[Union[an]]] }]; Print[a];

%Y Cf. A156831, A208951.

%K nonn

%O 1,3

%A _John W. Layman_, Mar 03 2012

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