  Final states with a vector boson and a hadronic jet allow one to infer the
Born-level kinematics of the underlying hard scattering process, thereby
probing the partonic structure of the colliding protons. At forward rapidities,
the parton collisions are highly asymmetric and resolve the parton
distributions at very large or very small momentum fractions, where they are
less well constrained by other processes. Using theory predictions accurate to
next-to-next-to-leading order (NNLO) in QCD for both $\mathrm{W}^{\pm}$ and
$\mathrm{Z}$ production in association with a jet at large rapidities at the
LHC, we perform a detailed phenomenological analysis of recent LHC
measurements. The increased theory precision allows us to clearly identify
specific kinematical regions where the description of the data is insufficient.
By constructing ratios and asymmetries of these cross sections, we aim to
identify possible origins of the deviations, and highlight the potential impact
of the data on improved determinations of parton distributions.
