  We review various combinatorial applications of field theoretical and matrix
model approaches to equilibrium statistical physics involving the enumeration
of fixed and random lattice model configurations. We show how the structures of
the underlying lattices, in particular their colorability properties, become
relevant when we consider hard-particles or fully-packed loop models on them.
We show how a careful back-and-forth application of results of two-dimensional
quantum gravity and matrix models allows to predict critical universality
classes and consequently exact asymptotics for various numbers, counting in
particular hard object configurations on fixed or random lattices and meanders.
