Lexicographic shellability of partial involutions

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 lexicographically shellable poset. We determine which subintervals of the Bruhat posets are Eulerian, and moreover, by studying certain embeddings of the symmetric groups and their involutions into rook matrices and partial involutions, respectively, we obtain new shelling orders on the corresponding order complexes.