Both the noncrossing partition/Kreweras lattice and parking
functions are well-studied objects in combinatorics. In 1997 Richard
Stanley found a bijection between the maximal chains in the lattice and
parking functions. We investigated what happens under the bijection
when we restrict the noncrossing partition lattice to certain induced
sublattices (certain bond lattices). I will describe all the lattices
involved, some enumerative properties, and recent work on the
h-polynomial. This is joint work over several projects with many
people.