$DIII$ clan combinatorics for the orthogonal Grassmannian

Borel subgroup orbits of the classical symmetric space $S{O}_{2n}\mathrm{/}G{L}_n$ are parametrized by $DIII$ $(n,n)$-clans. We group the clans into “sects” corresponding to Schubert cells of the orthogonal Grassmannian, thus providing a cell decomposition for $S{O}_{2n}\mathrm{/}G{L}_n$. We also compute a recurrence for the rank polynomial of the weak order poset on $DIII$ clans, and then describe explicit bijections between such clans, diagonally symmetric rook placements, certain pairs of minimally intersecting set partitions, and a class of weighted Delannoy paths. Clans of the largest sect are in bijection with fixed-point-free partial involutions.