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.