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

Offset

1,3

Links

Table of n, a(n) for n=1..8.

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

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

Crossrefs

Cf. A156831, A208951.

Sequence in context: A354150 A344782 A191240 * A091358 A091359 A059776

Adjacent sequences: A208949 A208950 A208951 * A208953 A208954 A208955

Keyword

nonn,more

Author

John W. Layman, Mar 03 2012

Status

approved