  Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted
plane trees are four classes of Catalan objects which carry a notion of type.
There exists a product formula which enumerates these objects according to
type. We define a notion of `connectivity' for these objects and prove an
analogous product formula which counts connected objects by type. Our proof of
this product formula is combinatorial and bijective. We extend this to a
product formula which counts objects with a fixed type and number of connected
components. We relate our product formulas to symmetric functions arising from
parking functions. We close by presenting an alternative proof of our product
formulas communicated to us by Christian Krattenthaler which uses generating
functions and Lagrange inversion.
