  The temperature-magnetic-field phase diagram of the mixed honeycomb
triangular lattice system K$_{2}$Mn$_{3}$(VO$_{4}$)$_{2}$CO$_{3}$ is
investigated by means of magnetization, heat capacity and neutron scattering
measurements. The results indicate that triangular and honeycomb magnetic
layers undergo sequential magnetic orderings and act as nearly independent
magnetic sublattices. The honeycomb sublattice orders at about 85 K in a
Ne\'{e}l-type antiferromagnetic structure, while the triangular sublattice
displays two consecutive ordered states at much lower temperatures, 3 K and 2.2
K. The ground state of the triangular sublattice consists of a planar `Y'
magnetic structure that emerges from an intermediate collinear `up-up-down'
state. Applied magnetic fields parallel or perpendicular to the $c$-axis induce
exotic ordered phases characterized by various spin-stacking sequences of
triangular layers that yield bilayer, three-layer or four-layer magnetic
superstructures. The observed superstructures cannot be explained in the
framework of quasi-classical theory based only on nearest-neighbor interlayer
coupling and point towards the presence of effective second-nearest-neighbor
interactions mediated by fluctuations of the magnetic moments in the honeycomb
sublattice.
