  This work concerns the study of properties 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 easily derived from lattice
theory for a subfamily coming from a two-sided ladder to a family where, as we
show, lattice theory no longer applies in any obvious way and includes
constructive proofs which may be useful in future study of these rings and
others.
