Eulerian numbers

The Eulerian numbers $A (n, k)$ are the number of permutations $w \in S_{n}$ with exactly $k$ descents

Examples

Let $n = 3$ . Then, $A (3, 0) = 1, A (3, 1) = 4, A (3, 2) = 1$ as we can see below

These are also the coefficients of the $n$ th Eulerian polynomial (proof here)
Let $x_{1}, x_{2}, \dots, x_{n}$ be independent random variables, uniformly distributed on $[0, 1]$ . Let $Y_{n} = ⌊ x_{1} + x_{2} + \dots + x_{n} ⌋$ a discrete random variable taking values ${0, \dots, n}$ . Then, $P (Y_{n} = k) = \frac{A (n, k)}{n!}$
- Geometrically, this looks like the unit hypercube being cut by the planes $x_{1} + \dots + x_{n} = k$ for each $k$ , and then asking for the volumes of these cuts.