Volumes of polytopes in hypercubes
Roland Bacher
Roland.Bacher at ujf-grenoble.fr
Sat Aug 19 12:41:28 CEST 2006
On Fri, Aug 18, 2006 at 09:13:00AM -0400, N. J. A. Sloane wrote:
> Roland said:
>
> It would be interesting to know them:
> The first value $n! Vol(P(0))=n! Vol(P(n-1))$ is always one.
> The triangle of these Volumes starts thus
>
> 1: 1
> 2: 1 1
> 3: 1 4 1
> 4: 1 11 11 1
> 5: 1 x y x 1
>
> where 2+2x+y=120 and y/120 is the volume of the convex hull
> P(2)=P(2)_5 of the 20 integral points with coordinates in {0,1}^5 and
> coordinate sum either 2 or 3. (Remark that this polytope is symmetric
> with respect to central symmetry in its barycenter 1/5(1,1,1,1,1),
> this holds of course for all "middle" polytopes P(n)_{2n+1}
> in odd dimension 2n+1).
>
> Me: Looks like the Eulerian numbers, A008292 !
>
> NJAS
A random computations for n=5,6 (choosing 100000 points uniformly
at random in [0,1] and computing the proportion of these points
in a given such polytope) seem to confirm the Eulerian numbers.
Roland
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