  Using first-principles calculations based on the density functional theory,
we show a strong strain dependence of magnetic order in
(LaMnO$_{3}$)$_n$/(SrMnO$_{3}$)$_{2n}$ (001) superlattices with $n=1,2$. The
epitaxial strain lifts the degeneracy of Mn $e_{g}$ orbitals, thus inducing an
inherent orbital order, which in turn strongly affects the ferromagnetic double
exchange of itinerant $e_{g}$ electrons, competing with the antiferromagnetic
superexchange of localized $t_{2g}$ electrons. For the case of tensile strain
induced by SrTiO$_3$ (001) substrate, we find that the ground state is A-type
antiferromagnetic and $d_{x^2-y^2}$ orbital ordered, which is in excellent
agreement with recent experiments [S. J. May {\it et al.}, Nature Materials
{\bf 8}, 892 (2009)]. Instead, for the case of compressive strain induced by
LaAlO$_3$ (001) substrate, we predict that the ground state is C-type
antiferromagnetic and $d_{3z^2-r^2}$ orbital ordered.
