 $Z_{R_{j} }^{\left(\pi '_{S} \right)} \left(\left(R_{i} ,\sigma _{k} \right)\right)$ cardinality of the coincidence set is $$\label{0)}  Z_{R_{j} }^{\left(\pi '_{S} \right)} \left(\left(R_{i} ,\sigma _{k} \right)\right)=Z_{R_{j} }^{\left(\pi _{S} \right)} \left(\left(R_{i} ,\sigma _{k} \right)\right)+\left|B\left(R_{i} \left(\pi _{S} \right),\sigma _{k} \right)\right|