The two-dimensional lattice Boltzmann method (LBM) was used to study the motion of two interacting particles with different densities (ρ_{1} and ρ_{2}) and diameters (d_{1} and d_{2}), which were placed in a vertical channel under gravity. Both the density ratio (λ=ρ_{2}/ρ_{1}) and diameter ratio (r=d_{2}/d_{1}) between the particles were considered. The transition boundaries between the regime where the particles settle separately and the regime where the particles interact are identified by λ_{max}(r) and λ_{min}(r); they exhibit excellent power-law relationships with r. A pattern of horizontal oscillatory motion (HOM), characterized by a structure with a large (but light) particle right above a small (but heavy) one and strong oscillations of both particles in the horizontal direction, was revealed for r∼0.3 at intermediate Reynolds numbers. The results indicate that the magnitude of oscillations decreases with λ, whereas the frequency displays the opposite trend. A sudden increase in the terminal velocity of particles is seen, illustrating a transition from the classical pattern of drafting, kissing, and tumbling to the HOM at a certain λ. Upon increasing λ, the pattern of HOM may bifurcate into a vertical steady state at low Re or small r. Furthermore, the effects of the confinement ratio and particle-to-fluid density ratio were also examined. The existence of a critical confinement ratio is observed, beyond which the particles interact in a different manner.