It is stated in this pdf that for $X_j \text{ iid},$ uniformly distributed on $[0,1],$
$$\dfrac{1}{n!} \left\langle n \atop k \right\rangle = P\left(\sum_{j=1}^{n}X_j\in[k,k+1]\right)$$
without further qualification. Despite multiple searches, I can't find a reference to the identity, or a proof anywhere. Where might I find an outline of a proof for this particular identity?
It is Theorem $\mathbf{2.1}$ in Kingo Kobayashi, Hajime Sato, Mamoru Hoshi, and Hiroyoshi Morita, Eulerian numbers revisited: Slices of hypercube [PDF]; their proof is short, about a page.