Special relations of coefficients of fractional parentage and partial dynamical symmetries in $j=9/2$ shells

The partial conservation of the seniority symmetry in $j=9/2$ shells has been discussed recently. In this work we show that the occurrence of the partial conservation of the seniority is the consequence of special properties of certain one-particle coefficients of fractional parentage. Further studies have also been done to pursue the similar special relations of the coefficients of fractional parentage in other single-$j$ shells with $j\le 15/2$.