Proper Partitions of a Polygon and k-Catalan Numbers

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.

• Type: Journal
• Year: 2008
• Where: ARSCOM
• Authors: Bruce E. Sagan
• Keywords: ARSCOM 2008, Conjectures, Polygon, Propp