  A search is presented for the production of a Higgs boson in association with
a single top quark, based on data collected in 2016 by the CMS experiment at
the LHC at a center-of-mass energy of 13 TeV, which corresponds to an
integrated luminosity of 35.9 fb$^{-1}$. The production cross section for this
process is highly sensitive to the absolute values of the top quark Yukawa
coupling, $y_t$, the Higgs boson coupling to vector bosons, $g_\mathrm{HVV}$,
and, uniquely, to their relative sign. Analyses using multilepton signatures,
targeting H $\to$ WW, H $\to$ $\tau\tau$, and H $\to$ ZZ decay modes, and
signatures with a single lepton and a $\mathrm{b\overline{b}}$ pair, targeting
the H $\to$ $\mathrm{b\overline{b}}$ decay, are combined with a
reinterpretation of a measurement in the H $\to$ $\gamma\gamma$ channel to
constrain $y_\mathrm{t}$. For a standard model-like value of $g_\mathrm{HVV}$,
the data favor positive values of $y_\mathrm{t}$ and exclude values of
$y_\mathrm{t}$ below about $-$0.9 $y_\mathrm{t}^\mathrm{SM}$.
