I have to describe the permutations which are both $312$-avoiding and $213$-avoiding, and then count how many there are of length $n$. So I found that when $n<4$ that the there are clearly no permutations which have both $213$ and $312$ patterns.
My problem is that this is not simply counting the permutations with $312$ and $213$ encoded into them. Because you can have something like $2413$ which has a subsequence $213$ and another subsequence $413$ which corresponds to the pattern $312$. I think it is possible to describe one type of permutation for any $n$ that avoids both but how is it possible to describe them all?