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We examine Borel subgroup orbits in the classical symmetric space of type CI, which are parametrized by skew symmetric (n, n)-clans. We describe bijections between such clans, certain weighted lattice paths, and pattern-avoiding signed involutions, and we give a cell decomposition of the symmetric space in terms of collections of clans called sects. The largest sect with a conjectural closure order is isomorphic (as a poset) to the Bruhat order on partial involutions.

Sects and Lattice Paths over the Lagrangian Grassmannian