Conditions on (d_1max, d_2max)

Embedded values

$d_{1\max }=1$ and $d_{2\max }=0$

$d_{1\max }=1$ or $d_{2\max }=0$

if (b1 == 0 and b2 == 0)

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

else if (b1 == 0 and b2 == 1)

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

else if (b1 == 1and b2 == 0)

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

else if (b1 == 1and b2 == 1

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

$d_{1\max }>1$ or $d_{2\max }<0$

if (b1 == 1)

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

else if (b1 == 0)

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

$d_{1\max }\in \left(-\infty ,-\text{1}\right]\cup \left[\text{2},\infty \right)$

$d_{2\max }=0$ and $d_{2\max }=0$

if (b1 == 1)

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

else if(b1 == 0)

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

$d_{1\max }\in \left(-\infty ,-\text{1}\right]\cup \left[\text{2},\infty \right)$

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