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We calculate the conductance through strongly correlated T-shaped molecular or quantum dot systems under the influence of phonons. The system is modelled by the extended Anderson-Holstein Hamiltonian. The finite-U mean-field slave boson approach is used to study many-body effects. Phonons influence both interference and correlations. Depending on the dot unperturbed energy and the strength of electron-phonon interaction, the system is occupied by a different number of electrons that effectively interact with each other repulsively or at
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