  We consider the problem of bounding the number of permutations $\sigma\in
S_n$ that avoid a fixed permutation $\pi\in S_k$ in specific indices given by a
$k$-uniform hypergraph $\Lambda$. We obtain relatively sharp bounds in the case
where $\Lambda$ is a random hypergraph, and find bounds in the case where
$\Lambda$ contains many large cliques. Along the way, we prove a
supersaturation version of F\"uredi-Hajnal, which may be of independent
interest.
