Properties of permutations with unknown pattern avoidance descriptions

Here's one idea. For every permutation $\pi$ of length $n$, there are $n^2+1$ permutation of length $n+1$ containing $\pi$. However, once you look at permutations of length $n+2$, this quantity depends on $\pi$. In their paper "Posets of matrices and permutations with forbidden subsequences", Ray and West gave a proof that for $\pi$ of length $n$ the number of permutations of length $n+2$ containing $\pi$ is $$ (n^4+2n^3+n^2+4n+4-2j)/2, $$ where $0\le j\le n-1$ depends on $\pi$. Perhaps you could give a description of this statistic in terms of patterns of $\pi$?

Derangements. More generally, properties that allow superexponentially many permutations.