There has been lots of studies on pattern-avoiding permutations. There is also some work done on permutations whose permutation matrix fit in a diagram shape, and then do pattern avoidance on this type of permutations. However, I am not aware of any work being done on the following type of questions:
Let $D$ be a (skew) Young diagram. We can say that $D$ avoids $D'$ if one cannot obtain $D'$ from $D$ by deleting boxes from $D$, and deleting empty rows and columns of $D$.
Note that this generalize the notion of permutation avoidance, if one allow more general shapes than just skew shapes.
Has this been considered before?
I realized that this type of avoidance appear naturally in some questions I am working on.