  A search for leptoquarks produced singly and in pairs in proton-proton
collisions is presented. We consider the leptoquark (LQ) to be a scalar
particle of charge -1/3$e$ coupling to a top quark plus a tau lepton
($\mathrm{t}\tau$) or a bottom quark plus a neutrino ($\mathrm{b}\nu$), or a
vector particle of charge +2/3$e$, coupling to $\mathrm{t}\nu$ or
$\mathrm{b}\tau$. These choices are motivated by models that can explain a
series of anomalies observed in the measurement of B meson decays. In this
analysis the signatures $\mathrm{t}\tau\nu\mathrm{b}$ and $\mathrm{t}\tau\nu$
are probed, using data recorded by the CMS experiment at the CERN LHC at
$\sqrt{s} =$ 13 TeV and that correspond to an integrated luminosity of 137
fb$^{-1}$. These signatures have not been previously explored in a dedicated
search. The data are found to be in agreement with the standard model
prediction. Lower limits at 95% confidence level are set on the LQ mass in the
range 0.98-1.73 TeV, depending on the LQ spin and its coupling $\lambda$ to a
lepton and a quark, and assuming equal branching fractions for the two LQ decay
modes considered. These are the most stringent constraints to date on the
existence of leptoquarks in this scenario.
