This thesis is comprised of four projects. The first project concerns q-analogues of the classical rook and hit numbers. These q-rook and q-hit numbers are defined in terms of the number of matrices with fixed rank and support contained in a fixed set over a finite field of size q. We show that their residues modulo small powers of q are polynomial in q. We find a formula for these polynomials and use it to prove a positivity result for the (q − 1) coefficient of the q-hit number. The second project concerns the signs of principal minors of a real symmetric matrix. These sign patterns are equivalent to uniform oriented Lagrangian matroids. We study their structure, symmetries, and asymptotics, proving that almost all of them are not representable by real symmetric matrices. We offer several conjectures and experimental results concerning representable sign patterns and the topology of their representation spaces. The third project concerns chromatic symmetric functions related to the Stanley-Stembridge conjecture. We show that the allowable coloring weights for indifference graphs of Dyck paths are the lattice points of a permutahedron P_λ, and we give a formula for the dominant weight λ. Furthermore, we prove that chromatic symmetric functions for abelian Dyck paths are Lorenzian. We extend our results on the Newton polytope to incomparability graphs of (3+1)-free posets, and give a number of conjectures and results stemming from our work, including results on the complexity of computing the coefficients and relations with the ζ map from diagonal harmonics. The fourth project concerns the regular and nilpotent Hessenberg variety of shape λ associated to the path graph. We give a combinatorial model in terms of rook walks for the cells in an affine paving of these varieties which is a special case of Tymoczko’s filling rule. We note a mysterious phenomenon in the Poincaré polynomial of the nilpotent Hessenberg variety of shape (n, n), and we present several questions with related experimental data.