We investigate the interplay between spin and orbital correlations in
monolayer and bilayer manganites using an effective spin-orbital t-J model
which treats explicitly the e_g orbital degrees of freedom coupled to classical
t_{2g} spins. Using finite clusters with periodic boundary conditions, the
orbital many-body problem is solved by exact diagonalization, either by
optimizing spin configuration at zero temperature, or by using classical
Monte-Carlo for the spin subsystem at finite temperature. In undoped
two-dimensional clusters, a complementary behavior of orbital and spin
correlations is found - the ferromagnetic spin order coexists with alternating
orbital order, while the antiferromagnetic spin order, triggered by t_{2g} spin
superexchange, coexists with ferro-orbital order. With finite crystal field
term, we introduce a realistic model for La_{1-x}Sr_{1+x}MnO_4, describing a
gradual change from predominantly out-of-plane 3z^2-r^2 to in-plane x^2-y^2
orbital occupation under increasing doping. The present electronic model is
sufficient to explain the stability of the CE phase in monolayer manganites at
doping x=0.5, and also yields the C-type antiferromagnetic phase found in
Nd_{1-x}Sr_{1+x}MnO_4 at high doping. Also in bilayer manganites magnetic
phases and the accompanying orbital order change with increasing doping. Here
the model predicts C-AF and G-AF phases at high doping x>0.75, as found
experimentally in La_{2-2x}Sr_{1+2x}Mn_2O_7.Comment: 23 pages, 21 figure