  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 $\Sigma_c\bar{D}_1$ state
with $I(J^P)=1/2(1/2^+)$, the $\Sigma_c\bar{D}_2^*$ 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.
