Check out A008292, the Eulerian numbers. You'll need to divide by n!. (Oh, you asked for <=k, so you'll want their partial sums -- looks like that's A179457.) Exercise: explain why :-) --Michael On Mon, Sep 3, 2012 at 8:51 AM, Erich Friedman <efriedma@stetson.edu> wrote:
greetings.
does anyone know the n-dimensional volume V(k,n) that satisfies:
0 <= x_i <= 1 (for 1 <= i <= n) and \sum_{i=1}^n x_i <= k (where k is some integer 0 <= k <= n)?
this is just the portion of the unit hypercube that is underneath the hyperplane x_1+...+x_n <= k.
or, if you're a probabilist like me, V(k,n) is the chance that the sum of k iid uniform [0,1] random variables is larger than the sum of (n-k) other iid uniform [0,1] random variables.
surely these are important enough to have been determined before, though i could not find these in the OEIS.
easy multiple integrals and symmetry show:
V(0,n) = 0 V(n,n) = 1 V(1,n) = 1/n! V(n-1,n) = 1 - 1/n! V(n/2,n) = 1/2 V(n-k,n) = 1 - V(k,n)
is there a general formula for V(k,n)? i couldn't even find a good way to calculate V(2,5)....
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