  A search is presented for the single production of vector-like quarks in
proton-proton collisions at $\sqrt{s} =$ 13 TeV. The data, corresponding to an
integrated luminosity of 35.9 fb$^{-1}$, were recorded with the CMS experiment
at the LHC. The analysis focuses on the vector-like quark decay into a top
quark and a W boson, with one muon or electron in the final state. The mass of
the vector-like quark candidate is reconstructed from hadronic jets, the
lepton, and the missing transverse momentum. Methods for the identification of
b quarks and of highly Lorentz boosted hadronically decaying top quarks and W
bosons are exploited in this search. No significant deviation from the standard
model background expectation is observed. Exclusion limits at 95% confidence
level are set on the product of the production cross section and branching
fraction as a function of the vector-like quark mass, which range from 0.3 to
0.03 pb for vector-like quark masses of 700 to 2000 GeV. Mass exclusion limits
up to 1660 GeV are obtained, depending on the vector-like quark type, coupling,
and decay width. These represent the most stringent exclusion limits for the
single production of vector-like quarks in this channel.
