  In this manuscript we study inclusion posets of Borel orbit closures on
(symmetric) matrices. In particular, we show that the Bruhat poset of partial
involutions is a lexicographiically shellable poset. Also, studying the
embeddings of symmetric groups and involutions into rooks and partial
involutions, respectively, we find new $EL$-labelings on permutations as well
as on involutions.
