Counting of Distinct Equivalence Classes of Circuits inPSL2, Z-Space

Abstract

Graham Higman was the first who studied the transitive actions of the extended modular group$\text{PGL}\left(2,\mathbb{Z}\right)$over$\text{PL}\left(F_q\right)=F_q{\displaystyle \cup \left\{\infty \right\}}$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${\left(x_1{x}_2\right)}^{{\pi }_1}{\left(x_1{x}_2^{-1}\right)}^{{\pi }_2}{\left(x_1{x}_2\right)}^{{\pi }_3},\dots ,{\left(x_1{x}_2^{-1}\right)}^{{\pi }_l}\in \text{PSL}\left(2,\mathbb{Z}\right)$, and it is denoted by$\left({\pi }_1,{\pi }_2,{\pi }_3,\dots ,{\pi }_l\right)$. In this study, we shall formulate combinatorial sequences and find the number of distinct equivalence classes of a length-6 circuit$\left({\pi }_1,{\pi }_2,{\pi }_3,{\pi }_4,{\pi }_5,{\pi }_6\right)$for a fixed number of triangle$\Delta$of class$\Pi$.