Bijections Between Colored Compositions, Dyck Paths, and Polygon Partitions

We give part-preserving bijections between three fundamental families of objects that serve as natural framework for many problems in enumerative combinatorics. Specifically, we consider compositions, Dyck paths, and partitions of a convex polygon, and identify suitable building blocks that are then appropriately decorated to achieve matching cardinalities. Our bijections are constructive and apply for the general case where the building blocks are allowed to come in different colors.