  The differential cross section and charge asymmetry for inclusive W boson
production at $\sqrt{s} =$ 13 TeV is measured for the two transverse
polarization states as a function of the W boson absolute rapidity. The
measurement uses events in which a W boson decays to a neutrino and either a
muon or an electron. The data sample of proton-proton collisions recorded with
the CMS detector at the LHC in 2016 corresponds to an integrated luminosity of
35.9 fb$^{-1}$. The differential cross section and its value normalized to the
total inclusive W boson production cross section are measured over the rapidity
range $|y_\mathrm{W}|$ $\lt$ 2.5. In addition to the total fiducial cross
section, the W boson double-differential cross section,
d$^2\sigma$/d$p^\ell_\mathrm{T}$d$|\eta|$ and the charge asymmetry are measured
as functions of the charged lepton transverse momentum and pseudorapidity. The
precision of these measurements is used to constrain the parton distribution
functions of the proton using the next-to-leading order NNPDF3.0 set.
