acase@tulane.edu / Let G be a reductive group, B be a Borel subgroup, and let K be a symmetric subgroup of G. The study of B orbits in a symmetric variety G/K or, equivalently, the study of K orbits in a flag variety G/B has importance in the study of Harish-Chandra modules; it comes with many interesting Schubert calculus problems. Although this subject is very well studied, it still has many open problems from combinatorial point of view. The most basic question that we want to be able to answer is that how many B orbits there are in G/K. In this thesis, we study the enumeration problem of Borel orbits in the case of classical symmetric varieties. We give explicit formulas for the numbers of Borel orbits on symmetric varieties for each case and determine the generating functions of these numbers. We also explore relations to lattice path enumeration for some cases. In type A, we realize that Borel orbits are parameterized by the lattice paths in a pxq grid moving by only horizontal, vertical and diagonal steps weighted by an appropriate statistic. We provide extended results for type C as well. We also present various t-analogues of the rank generating function for the inclusion poset of Borel orbit closures in type A. / 1 / Ozlem Ugurlu

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