Let $A(m,n)$ denote the Eulerian numbers. I'm looking for a simple combinatorial proof of the following fact.

Fact.If $p$ is prime and $0\le k < p-1$, then $A(p-1,k) \equiv 1 \pmod{p}$.

The closest thing I'm aware of is an argument of S. Tanimoto, An operation on permutations and its application to Eulerian numbers, *European Journal of Combinatorics* **22** (2001), 569–576 that can be adapted to give a rather complicated proof, but I'm hoping for something simpler and more direct.