Inspired by the observations of three $P_c$ states, we systematically investigate interactions between an $S$-wave charmed baryon $mathcal{B}_{c}^{(*)}=Lambda_c/Sigma_c/Sigma_c^{*}$ and an anticharmed meson $bar T=bar D_1/bar D_2^*$ with the one-pion-exchange potential model and the one-boson-exchange potential model, and search for possible new types of $P_c$ states with the structures of $mathcal{B}_{c}^{(*)}bar T$. Both $S$-$D$ wave mixing and coupled channel effects are considered. Our results suggest that in some $mathcal{B}_{c}^{(*)}bar T$ systems there are ideal candidates of new types of $P_c$ states, i.e., the $Sigma_cbar{D}_1$ state with $I(J^P)=1/2(1/2^+)$, the $Sigma_cbar{D}_2^*$ state with $I(J^P)=1/2(3/2^+)$, the $Sigma_c^*bar{D}_1$ state with $I(J^P)=1/2(1/2^+)$, and the $Sigma_c^*bar{D}_2^*$ states with $I(J^P)=1/2(1/2^+, 3/2^+)$, and we suggest that these predicted new types of $P_c$ states can be detected in the process $Lambda_b^0 to psi(2S) p pi^{-}$. Meanwhile, we also extend our study to the interactions between an $S$-wave charmed baryon and a charmed meson in a $T$ doublet, and we predict a series of double-charm molecular pentaquarks.
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