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.

  • $\begingroup$ Okay, another try: is this notion of $D$ containing $D'$ equivalent to $P_D$ containing an induced copy of $P_{D'}$, where $P_D$ is the poset associated to the diagram $D$? $\endgroup$ – Sam Hopkins Apr 8 '15 at 16:33
  • $\begingroup$ I rather think of it as matrices with boxes or non-boxes. If you can delete rows and columns of $D$, such that every position with a box in $D'$ has a box in $D$, then $D$ contains $D'$. Note that avoiding $D_1 \cup D_2$ and avoiding $D_2 \cup D_1$ is two different types of avoidance, where $\cup$ means "put $D_1$ up right, and $D_2$ down bottom". $\endgroup$ – Per Alexandersson Apr 8 '15 at 16:45

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