  A search is performed for a heavy particle decaying into different-flavor,
dilepton pairs ($e\mu$, $e\tau$ or $\mu\tau$), using 36.1 fb$^{-1}$ of
proton-proton collision data at $\sqrt{s}=13$ TeV collected in 2015-2016 by the
ATLAS detector at the Large Hadron Collider. No excesses over the Standard
Model predictions are observed. Bayesian lower limits at the 95% credibility
level are placed on the mass of a $Z^{\prime}$ boson, the mass of a
supersymmetric $\tau$-sneutrino, and on the threshold mass for quantum
black-hole production. For the $Z^{\prime}$ and sneutrino models, upper
cross-section limits are converted to upper limits on couplings, which are
compared with similar limits from low-energy experiments and which are more
stringent for the $e\tau$ and $\mu\tau$ modes
