Number of Distinct Fragments in Coset Diagrams for $PSL\left(2,\mathbb{Z}\right)$

Coset diagrams [1, 2] are used to demonstrate the graphical representation of the action of the extended modular group $\mathrm{P}\mathrm{G}\mathrm{L}\left(2,\mathbb{Z}\right)$ over $\mathrm{P}\mathrm{L}\left(F_q\right)=F_q\bigcup \left\{\infty \right\}$ . In these sorts of graphs, a closed path of edges and triangles is known as a circuit, and a fragment is emerged by the connection of two or more circuits. The coset diagram evolves through the joining of these fragments. If one vertex of the circuit is fixed by ${\left(ax\right)}^{{\rho }_1}{\left(a{x}^{-1}\right)}^{{\rho }_2}{\left(ax\right)}^{{\rho }_3}\cdots {\left(a{x}^{-1}\right)}^{{\rho }_k}\in \mathrm{P}\mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right)$ , then this circuit is termed to be a length – $k$ circuit, denoted by $\left({\rho }_1,{\rho }_2,{\rho }_3,\cdots ,{\rho }_k\right)$ . In this study, we consider two circuits of length $-6$ as ${\mathrm{\Omega }}_1=\left({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\alpha }_6\right)$ and ${\mathrm{\Omega }}_2=\left({\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4,{\beta }_5,{\beta }_6\right)$ with the vertical axis of symmetry that is ${\alpha }_2={\alpha }_6,{\alpha }_3={\alpha }_5$ and ${\beta }_2={\beta }_6,{\beta }_3={\beta }_5$ . It is supposed that $\mathrm{\Omega }$ is a fragment formed by joining ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ at a certain point. The condition for existence of a fragment is given in [3] in the form of a polynomial in $\mathbb{Z}\left[z\right]$ . If we change the pair of vertices and connect them, then the resulting fragment and the fragment $\mathrm{\Omega }$ may coincide. In this article, we find the total number of distinct fragments by joining all the vertices of ${\mathrm{\Omega }}_1$ with the vertices of ${\mathrm{\Omega }}_2$ provided the condition ${\beta }_4<{\beta }_3<{\beta }_2<{\beta }_1<{\alpha }_4<{\alpha }_3<{\alpha }_2<{\alpha }_1$ .