  The first implementation of fragmentation in a numerical code for the
computation of cross sections at next-to-next-to-leading order in QCD has
recently been completed. I will present some results of the first application
of this new framework to the production of top-quark pairs at the LHC in
association with a bottom-flavoured hadron. Additionally, I will present an
extended version of this calculation, which includes the decay of the
$B$-hadron to a $J/\psi$ meson or a muon.
