Let $G=GL(n)$, $B$ the subgroup of upper-triangular matrices, and
$K=GL(p) \times GL(q)$ where $p+q=n$. The group $K$ acts with finitely many orbits on the flag variety $G/B$, and one can study the closures of $K$-orbits just as one studies Schubert varieties, which are the closures of $B$-orbits. The set of $K$-orbits is parameterized by combinatorial objects known as $(p,q)$-clans. I will explain an older theorem relating interval pattern avoidance on permutations and singularities of Schubert varieties and how to extend this relationship to $(p,q)$-clans and $K$-orbit closures.
This is joint work with Ben Wyser and Alexander Yong.