Alternatively, the following two matrices define an isomorphic Leonard pair on $V$: $$\label{split2}
\left(\begin{array}{ccccc} \theta_d \\ 1 & \theta_{d-1} \\ & 1 & \theta_{d-2} \\
& & \ddots & \ddots \\ & & & 1 & \theta_0 \end{array}\right),\qquad
\left(\begin{array}{ccccc} \theta^*_0 & \phi_1 \\ & \theta^*_1 &
\phi_2 \\ & & \theta^*_2 & \ddots \\
& & & \ddots  & \phi_d \\ & & & & \theta^*_d \end{array}\right).$$

Conversely, if $(A,A^*)$ is a Leonard pair on $V$, there exists [@terwgen Section 21] a basis for $V$ with respect to which the matrices for $A$, $A^*$ have the bidiagonal forms in (\[split1\]), respectively. There exists another basis for $V$ with respect to which the matrices for $A$, $A^*$ have the bidiagonal forms in (\[split2\]), respectively, with the same scalars $\theta_0,\theta_1,\ldots,\theta_d;
\theta^*_0,\theta^*_1,\ldots,\theta^*_d$. Then the following 4 sequences are parameter arrays of diameter $d$: $$\begin{aligned}
 \label{parray1}
(\theta_0,\theta_1,\ldots,\theta_d;\;\theta_0^*,\theta_1^*,\ldots,\theta_d^*;\;
\varphi_1,\ldots,\varphi_d;\;\phi_1,\ldots,\phi_d),\\
$$$$\begin{aligned}
\label{parray2}
(\theta_0,\theta_1,\ldots,\theta_d;\;\theta_d^*,\ldots,\theta_{1}^*,\theta_0^*;\;
\phi_d,\ldots,\phi_1;\;\varphi_d,\ldots,\varphi_1),\\ \label{parray3}
(\theta_d,\ldots,\theta_{1},\theta_0;\;\theta_0^*,\theta_1^*,\ldots,\theta_d^*;\;
\phi_1,\ldots,\phi_d;\;\varphi_1,\ldots,\varphi_d),\\ \label{parray9}
(\theta_d,\ldots,\theta_{1},\theta_0;\;\theta_d^*,\ldots,\theta_{1}^*,\theta_0^*;\;
\varphi_d,\ldots,\varphi_1;\;\phi_d,\ldots,\phi_1).\end{aligned}$$ If we apply to any of these parameter arrays the construction above, we get back a Leonard pair isomorphic to $(A,A^*)$. These are all parameter arrays which correspond to $(A,A^*)$ in this way.

The parameter arrays in (\[parray1\])–(\[parray9\]) are related by permutations. The permutation group is 