Quantum numbers of heavy neutrinos, tri-bi-maximal mixing through double seesaw with permutation symmetry, and comment on ${\theta }_{\mathrm{s}\mathrm{o}\mathrm{l}}+{\theta }_c\simeq \frac{\pi }{4}$

Using the family symmetry, in the neutrino mass matrix we remove the Yukawa coupling (arising in the Dirac type mass between the heavy neutrinos and light lepton doublets) dependence in the double seesaw mechanism so that it is directly proportional to the mass matrix $m^{(nn)}$ of heavy Majorana neutrinos. The family symmetry is supposed to be broken spontaneously at high energy scale so that the neutrino mass matrix is given by the family symmetry at high energy scale. With the permutation symmetry $S_3$, we note a variety of possible mass hierarchies arising distinctly in neutrinos, charged leptons, $Q_{\mathrm{e}\mathrm{m}}=-\frac{1}{3}$ quarks, and $Q_{\mathrm{e}\mathrm{m}}=\frac{2}{3}$ quarks. Distinguishing these hierarchies, we obtain a relation between the CKM angles and the MNS angles. Finally, we comment on the approximate relation $\theta_{\rm sol}+\theta_c\simeq \frac{\pi}{4}$.