  Three of the most significant measured deviations from standard model
predictions, the enhanced decay rate for B $\to$ D$^{(*)}\tau\nu$, hints of
lepton universality violation in B $\to$ K$^{(*)}\ell\ell$ decays, and the
anomalous magnetic moment of the muon, can be explained by the existence of
leptoquarks (LQs) with large couplings to third-generation quarks and masses at
the TeV scale. The existence of these states can be probed at the LHC in high
energy proton-proton collisions. A novel search is presented for pair
production of LQs coupled to a top quark and a muon using data at a
center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of
35.9 fb$^{-1}$, recorded by the CMS experiment. No deviation from the standard
model prediction has been observed and scalar LQs decaying exclusively into
t$\mu$ are excluded up to masses of 1420 GeV. The results of this search are
combined with those from previous searches for LQ decays into t$\tau$ and
b$\nu$, which excluded scalar LQs below masses of 900 and 1080 GeV. Vector LQs
are excluded up to masses of 1190 GeV for all possible combinations of
branching fractions to t$\mu$, t$\tau$ and b$\nu$. With this analysis, all
relevant couplings of LQs with an electric charge of -1/3 to third-generation
quarks are probed for the first time.
