Counting of Distinct Equivalence Classes of Circuits in PSL2, Z-Space

Graham Higman was the first who studied the transitive actions of the extended modular group PGL2, Z over PLFq=Fq∪∞ graphically and named it as coset diagram. In these sorts of graphs, a closed path of edges and triangles is known as a circuit. Coset diagrams evolve through the joining of these circuits. In a coset diagram, a circuit is termed as a length-l circuit if its one vertex is fixed by x1x2π1x1x2−1π2x1x2π3,…,x1x2−1πl∈PSL2, Z, and it is denoted by π1,π2,π3,…,πl. In this study, we shall formulate combinatorial sequences and find the number of distinct equivalence classes of a length-6 circuit π1,π2,π3,π4,π5,π6 for a fixed number of triangle Δ of class Π.