1. DIII clan combinatorics for the orthogonal Grassmannian.
    Australasian Journal of Combinatorics, Volume 79(1) (2021), Pages 55–-86.
    https://arxiv.org/pdf/1907.08875.pdf. (2020)

    Abstract: Borel subgroup orbits of the classical symmetric space SO(2n)/GL(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 SO(2n)/GL(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.

  2. Sects and lattice paths over the Lagrangian Grassmannian.
    The Electronic Journal of Combinatorics, Volume 27, Issue 1, P1.51 (2020). https://doi.org/10.37236/8664

    Abstract: 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.

  3. The genesis of involutions (polarizations and lattice paths).
    Discrete Mathematics, Volume 342, Issue 1, Pages 201-216 (2019). https://doi.org/10.1016/j.disc.2018.09.026

    Abstract: The number of Borel orbits in polarizations (the symmetric variety SL(n)/S(GL(p) x GL(q))) is analyzed, various (bivariate) generating functions are found. Relations to lattice path combinatorics are explored.

  4. Counting Borel Orbits in Symmetric Spaces of Types BI and CII .
    Arnold Mathematical Journal, volume 4, pages 213–250 (2018). http://DOI:10.1007/s40598-018-0092-3.

    Abstract: This is a continuation of our combinatorial program on the enumeration of Borel orbits in symmetric varieties of classical types. Here, we determine the generating series the numbers of Borel orbits in SO(2n+1)/S(O(2p) x O(2q+1)) (type BI) and in Sp(n)/Sp(p) x Sp(q)
    (type CII). In addition, we explore relations to lattice path enumeration.