Eulerian numbers revisited: Slices of hypercube

In this talk, we provide a simple proof on an interesting equality connecting the number of permutations of 1, ..., n with k runs, i.e., Eulerian numbers to the volumes of slices between k-1 and k of the n-dimensional hypercube along the diagonal axis. The proof is simple and elegant, but the detail structures in the problem are left to be unclear. In order to get more information on this problem, we give the second proof relied on the direct calculation of the related numbers and the volumes. By computing conditional probabilities with respect to slices, we can obtain the known recurrence relation on Eulerian numbers.