Volumes of polytopes in hypercubes

[Roland Bacher]

In fact, it is already in the OEIS:

In Formula for A8292, there is the entry:

 3. Let Cn be the unit cube in R^n with vertices (e_1, e_2, ..., e_n) where each e_i is 0 or 1 and all 2^n combinations are used. Then T(i, n)/n! is the volume of Cn between the hyperplanes x_1 + x_2 + ... + x_n = i and x_1 + x_2 + ... + x_n = i+1. Hence T(i, n)/n! is the probability that i <= X_1 + X_2 + ... + X_n < i+1 where the X_j are independent uniform [0, 1] distributions. - See Ehrenborg & Readdy reference.

I should have looked better,  Roland


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