  Let P be a polygon whose vertices have been colored (labeled) cyclically with
the numbers 1,2,...,c. Motivated by conjectures of Propp, we are led to
consider partitions of P into k-gons which are proper in the sense that each
k-gon contains all c colors on its vertices. Counting the number of proper
partitions involves a generalization of the k-Catalan numbers. We also show
that in certain cases, any proper partition can be obtained from another by a
sequence of moves called flips.
