Suitable vortex dipole pairs (modons) in eastward flow as well as monopole vortices in β-plane channel flow are characterized systematically in appropriate parameter regimes as the most-probable large-scale mean-field states predicted from a recent statistical theory (Turkington, 1998); this theory utilizes only a few conserved quantities involving energy, circulation, potential vorticity extrema, and the mean potential vorticity magnitude. The large-scale coherent structures emerge systematically from the statistical theory through maximization of a suitable coarse-grained entropy functional subject to the constraints imposed by these few conserved quantities. An accurate numerical procedure is developed here to study these states. For dilute PV theory, the most-probable large-scale coherent structures in eastward mean flows with nonzero β-effect are either dipolar vortex streets or zonal shear flows. The transition boundary of the predicted large-scale coherent structures between coherent vortices and zonal shear flows is related to a generalized Rhines' scale as the β-effect and energy are varied. The role of symmetry groups in the potential vorticity is emphasized here. In particular, in some parameter regimes the most-probable state within a given symmetry group of dipole pairs is not necessarily the most-probable large-scale coherent structure when the symmetry is broken.