ardinality of the coincidence sets at a given $\pi _{S} $, as $$\label{ZEqnNum000000}  \left|Z_{R_{j} } \left(\pi _{S} \right)\right|=\sum _{i,k}Z_{R_{j} }^{\left(\pi _{S} \right)} \left(\left(R_{i} ,\sigma _{k} \right)\right) ,$$ 