The decay $overline{B}_s^0 rightarrow psi(2S) K^+ pi^-$ is observed using a data set corresponding to an integrated luminosity of $3.0fb^{-1}$ collected by the LHCb experiment in $pp$ collisions at centre-of-mass energies of 7 and 8 TeV. The branching fraction relative to the $B^0rightarrow psi(2S) K^+ pi^-$ decay mode is measured to be begin{equation} frac{{cal B}(overline{B}^0_s rightarrow psi(2S) K^+ pi^-)}{{cal B}(B^0 rightarrow psi(2S) K^+ pi^-)} = 5.38 pm 0.36 (stat) pm 0.22 (syst) pm 0.31 , (f_s/f_d) , %, onumber end{equation} where $f_s/f_d$ indicates the uncertainty due to the ratio of probabilities for a $b$ quark to hadronise into a $B_s^0$ or $B^0$ meson. Using an amplitude analysis, the fraction of decays proceeding via an intermediate $K^*(892)^0$ meson is measured to be $0.645 pm 0.049 (stat) pm 0.049 (syst)$ and its longitudinal polarisation fraction is $0.524 pm 0.056 (stat) pm 0.029 (syst)$. The relative branching fraction for this component is determined to be begin{equation} frac{{cal B}(overline{B}^0_s rightarrow psi(2S) K^*(892)^0)}{{cal B}(B^0 rightarrow psi(2S) K^*(892)^0)} = 5.58 pm 0.57 (stat) pm 0.40 (syst) pm 0.32 , (f_s/f_d) , %. onumber end{equation} In addition, the mass splitting between the $B_s^0$ and $B^0$ mesons is measured as begin{equation} M(B^0_s) - M(B^0) = 87.45 pm 0.44 (stat) pm 0.07 (syst) MeV/c^2. onumber end{equation}
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