  The production of a $W$-boson with a charm quark jet provides a highly
sensitive probe of the strange quark distribution in the proton. Employing a
novel flavour dressing procedure to define charm quark jets, we compute
$W$+charm-jet production up to next-to-next-to-leading order (NNLO) in QCD. We
study the perturbative stability of production cross sections with same-sign
and opposite-sign charge combinations for the $W$ boson and the charm jet. A
detailed breakdown according to different partonic initial states allows us to
identify particularly suitable observables for the study of the quark parton
distributions of different flavours.
