The fusion and transfer induced fission reaction $^{9}$Be($^{238}$U,~f) with 6.2 MeV/u beam energy, using a unique setup consisting of AGATA, VAMOS++ and EXOGAM detectors, was used to populate through the fission process and study the neutron-rich $^{119,121}$In isotopes. This setup enabled the prompt-delayed $gamma$-ray spectroscopy of isotopes in the time range of $100~rm{ns} - 200~murm{s}$. In the odd-$A$ $^{119,121}$In isotopes, indications of a short half-life $19/2^{-}$ isomeric state, in addition to the previously known $25/2^{+}$ isomeric state, were observed from the present data. Further, new prompt transitions above the $25/2^{+}$ isomer in $^{121}$In were identified along with reevaluation of its half-life. The experimental data were compared with the theoretical results obtained in the framework of large-scale shell-model calculations in a restricted model space. The $langle pi g_{9/2} u h_{11/2};I arrowvert hat{mathcal{H}}arrowvert pi g_{9/2} u h_{11/2};Irangle$ two-body matrix elements of residual interaction were modified to explain the excitation energies and the $B(E2)$ transition probabilities in the neutron-rich In isotopes. The (i) decreasing trend of $E(29/2^{+}) - E(25/2^{+})$ in odd-In (with dominant configuration $pi g_{9/2}^{-1} u h_{11/2}^{-2}$ and maximum aligned spin of $29/2^{+}$) and (ii) increasing trend of $E(27/2^{+}) - E(23/2^{+})$ in odd-Sb (with dominant configuration $pi g_{7/2}^{+1} u h_{11/2}^{-2}$ and maximum aligned spin of $27/2^{+}$) with increasing neutron number could be understood as a consequence of hole-hole and particle-hole interactions, respectively.
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