  We consider a one-dimensional gas of $N$ charged particles confined by an
external harmonic potential and interacting via the one-dimensional Coulomb
potential. For this system we show that in equilibrium the charges settle, on
an average, uniformly and symmetrically on a finite region centred around the
origin. We study the statistics of the position of the rightmost particle
$x_{\max}$ and show that the limiting distribution describing its typical
fluctuations is different from the Tracy-Widom distribution found in the
one-dimensional log-gas. We also compute the large deviation functions which
characterise the atypical fluctuations of $x_{\max}$ far away from its mean
value. In addition, we study the gap between the two rightmost particles as
well as the index $N_+$, i.e., the number of particles on the positive
semi-axis. We compute the limiting distributions associated to the typical
fluctuations of these observables as well as the corresponding large deviation
functions. We provide numerical supports to our analytical predictions. Part of
these results were announced in a recent Letter, Phys. Rev. Lett. 119, 060601
(2017).
