  A search for heavy Majorana neutrinos in scattering of same-sign $W$ boson
pairs in proton-proton collisions at $\sqrt{s}= 13$ TeV at the LHC is reported.
The dataset used corresponds to an integrated luminosity of 140 fb$^{-1}$,
collected with the ATLAS detector during 2015-2018. The search is performed in
final states including a same-sign $ee$ or $e\mu$ pair and at least two jets
with large invariant mass and a large rapidity difference. No significant
excess of events with respect to the Standard Model background predictions is
observed. The results are interpreted in a benchmark scenario of the
Phenomenological Type-I Seesaw model. New constraints are set on the values of
the $\vert V_{e N} \vert^{2}$ and $\vert V_{e N} V^{*}_{\mu N} \vert$
parameters for heavy Majorana neutrino masses between 50 GeV and 20 TeV, where
$V_{\ell N}$ is the matrix element describing the mixing of the heavy Majorana
neutrino mass eigenstate with the Standard Model neutrino of flavour $\ell = e,
\mu$. The sensitivity to the Weinberg operator is investigated and constraints
on the effective $ee$ and $e\mu$ Majorana neutrino masses are reported. The
statistical combination of the $ee$ and $e\mu$ channels with the previously
published $\mu\mu$ channel is performed.
