We consider permutations which are $1324$ and $\overline{2143}$ avoiding, where $\overline{2143}$ avoiding means that it is $2143$ avoiding with the additional Bruhat restriction $\{2\leftrightarrow3\}$. In particular, for every permutation $\pi$ we will construct a linear map $L_\pi$ and a labeled graph $G_\pi$ and will show that the following three conditions are equivalent: $\pi$ is $1324$ and $\overline{2143}$ avoiding; $L_\pi$ is onto; $G_\pi$ is a forest. If time allows we will give a constructive proof showing that the $n$th Catalan number gives a lower bound for the number of such permutations in $S_n$. \vskip .1in \noindent This answers a conjecture of Woo and Yong, which shows that such permutations characterize which Schubert varieties are factorial.