The production of the hidden-charm pentaquarks $P_{c}$ via pion-induced reaction on a proton target is investigated within an effective Lagrangian approach. Three experimentally observed states, $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$, are considered in the calculation, and the Reggeized $t$-channel meson exchange is considered as main background for the reaction $pi ^{-}prightarrow J/psi n$. The numerical results show that the experimental data of the total cross section of the reaction $pi^{-}prightarrow J/psi n$ at $Wsimeq 5$ GeV can be well explained by contribution of the Reggeized $t$ channel with reasonable cutoff. If the branching ratios $Br[P_{c}rightarrow J/psi N]simeq 3%$ and $Br[P_{c}rightarrow pi N]simeq 0.05%$ are taken, the average value of the cross section from the $P_{c}(4312)$ contribution is about 1.2 nb/100 MeV, which is consistent with existing rude data at near-threshold energies. The results indicate that the branching ratios of the $P_{c}$ states to the $J/psi N$ and $pi N$ should be small. The shape of differential cross sections shows that the Reggeized $t$-channel provides a sharp increase at extreme forward angles, while the differential cross sections from the $P_{c}$ states contributions are relatively flat. High-precision experimental measurements on the reaction $pi ^{-}prightarrow J/psi n$ at near-threshold energies are suggested to confirm the LHCb hidden-charm pentaquarks as genuine states, and such experiments are also helpful to understand the origin of these resonance structures.
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