  The production of $\Upsilon(nS)$ mesons ($n=1,2,3$) in $p$Pb and Pb$p$
collisions at a centre-of-mass energy per nucleon pair $\sqrt{s_{NN}}=8.16$ TeV
is measured by the LHCb experiment, using a data sample corresponding to an
integrated luminosity of 31.8 nb$^{-1}$. The $\Upsilon(nS)$ mesons are
reconstructed through their decays into two opposite-sign muons. The
measurements comprise the differential production cross-sections of the
$\Upsilon(1S)$ and $\Upsilon(2S)$ states, their forward-to-backward ratios and
nuclear modification factors, performed as a function of the transverse
momentum \pt and rapidity in the nucleon-nucleon centre-of-mass frame $y^*$ of
the $\Upsilon(nS)$ states, in the kinematic range $p_{\rm{T}}<25$ GeV/$c$ and
$1.5<y^*<4.0$ ($-5.0<y^*<-2.5$) for $p$Pb (Pb$p$) collisions. In addition,
production cross-sections for $\Upsilon(3S)$ are measured integrated over phase
space and the production ratios between all three $\Upsilon(nS)$ states are
determined. The measurements are compared to theoretical predictions and
suppressions for quarkonium in $p$Pb collisions are observed.
