Recall that a permutation $\pi$ is vexillary if its Rothe diagram can be transformed to a Young diagram shape via some permutation of rows and columns. Lascoux and Schützenberger showed that this is equivalent to $\pi$ being 2143-avoiding. Let us call a permutation $\pi$ "skew vexillary" if its Rothe diagram can be transformed to a skew Young diagram shape via some permutation of rows and columns. (This is not such an unreasonable thing to consider: if $\pi$ is skew vexillary then its Stanley symmetric function $F_{\pi}$ is equal to a skew Schur function- and I don't know any examples of $\pi$ with $F_{\pi}$ equal to a skew Schur function which are not skew vexillary.)

I do not know of a pattern avoidance criteria describing the set of skew vexillary permutations. As mentioned in Remark 3.8 of https://arxiv.org/abs/1811.02404, I doubt there is a description using classical pattern avoidance because all permutations in $S_5$ are skew vexillary, but only 682 of the 720 patterns in $S_6$ are skew vexillary, so it would have to involve the avoidance of at least 38 patterns of length 6. On the other hand, perhaps this condition can be described by one of the generalizations of classical pattern avoidance.