This project concerns the classification and study of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain ladder-like structures). We determine a linear system of parameters for each ring and explicitly determine the Hilbert series for the resulting Artinian reduction. As corollaries, we obtain the multiplicity and regularity of the original rings. This work extends results known for a subfamily coming from a two-sided ladder and includes constructive proofs which may be useful in future study of these rings and others. We also develop explicit elements in the Priddy complex which correspond via known isomorphisms to Tate variables in the acyclic closure of the residue field over the localization of our rings at their homogeneous maximal ideals
