Conditions on ( ${d^{\prime }}_{1\max }$ , ${d^{\prime }}_{2\max }$ )

Extracted values

${d^{\prime }}_{1\max }=1$ and ${d^{\prime }}_{2\max }=0$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 0 and b2 = 0

${d^{\prime }}_{1\max }=0$ and ${d^{\prime }}_{2\max }=1$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 0 and b2 = 0

${d^{\prime }}_{1\max }=0$ and ${d^{\prime }}_{2\max }=0$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 0 and b2 = 0

${d^{\prime }}_{1\max }=1$ and ${d^{\prime }}_{2\max }=1$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 0 and b2 = 0

${d^{\prime }}_{2\max }=2$ and ${d^{\prime }}_{2\max }=2$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}-1$

b1 = 1 and b2 = 1

${d^{\prime }}_{1\max }=2$ and ${d^{\prime }}_{2\max }=-1$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}-1$

b1 = 1 and b2 = 1

${d^{\prime }}_{1\max }=-1$ and ${d^{\prime }}_{2\max }=-1$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}-1$

b1 = 1 and b2 = 1

${d^{\prime }}_{1\max }=-1$ and ${d^{\prime }}_{2\max }=2$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}-1$

b1 = 1 and b2 = 1

${d^{\prime }}_{1\max }=2$ and ${d^{\prime }}_{2\max }=1$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)}+1,P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 1 and b2 = 0

${d^{\prime }}_{1\max }=2$ and ${d^{\prime }}_{2\max }=0$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)}+1,P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 1 and b2 = 0

${d^{\prime }}_{1\max }=-1$ and ${d^{\prime }}_{2\max }=1$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)}+1,P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 1 and b2 = 0

${d^{\prime }}_{1\max }=-1$ and ${d^{\prime }}_{2\max }=0$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)}+1,P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 1 and b2 = 0

${d^{\prime }}_{1\max }=1$ and ${d^{\prime }}_{2\max }=-1$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)}+1,P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 0 and b2 = 1

${d^{\prime }}_{1\max }=1$ and ${d^{\prime }}_{2\max }=2$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)}+1,P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 0 and b2 = 1

${d^{\prime }}_{1\max }=0$ and ${d^{\prime }}_{2\max }=2$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)}+1,P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 0 and b2 = 1

${d^{\prime }}_{1\max }=0$ and ${d^{\prime }}_{2\max }=-1$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)}+1,P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 0 and b2 = 1

${d^{\prime }}_{1\max }=1$ and ${d^{\prime }}_{2\max }\in \left(-\infty ,-\text{1}\right)\cup \left(\text{2},\infty \right)$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)}+1,P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 0

${d^{\prime }}_{1\max }=0$ and ${d^{\prime }}_{2\max }\in \left(-\infty ,-\text{1}\right)\cup \left(\text{2},\infty \right)$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)}+1,P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}$

b1 = 0

${d^{\prime }}_{1\max }=2$ and ${d^{\prime }}_{2\max }\in \left(-\infty ,-\text{1}\right)\cup \left(\text{2},\infty \right)$

$P_{\sigma \left(2\right)}={P^{\prime }}_{\sigma \left(2\right)},P_{\sigma \left(1\right)}={P^{\prime }}_{\sigma \left(1\right)},P_{\sigma \left(3\right)}={P^{\prime }}_{\sigma \left(3\right)}-1$

b1 = 1