Direct charmonium productions and prompt J/ψ productions are studied for the central Au-Au collisions at RHIC and LHC energies, where the equilibrating partonic system is succeeded with hadronic matter in chemical nonequilibrium. Charmonium dissociation cross sections by mesons in the pseudoscalar octet and vector nonet are calculated with a QCD-based potential developed from the Buchmüller-Tye potential. A prescription for adopting c c radius dependence of c-c relativemotion wave functions provides c c dissociation cross sections by a gluon and mesons as functions of the radius. Deconfinement enhancement originating from the partonic system appears obviously in the prompt J/ψ number distributions and direct charmonium number distributions in the rapidity region -1.5 < y < 1.5 at p T = 4 GeV but the direct ψ at LHC energy. Such deconfinement enhancement is also found in the ratios of prompt J/ψ and direct charmonium number distributions to the initially yielded number distributions for Au-Au collisions or to nucleon-nucleon collisions at the RHIC energy. Twists in the prompt J/ψ and direct charmonium number distributions and the ratios in -1.5 < y < 1.5 show the existence of partonic system at the LHC energy.