  For any positive integers $a$ and $b$, we enumerate all colored partitions
made by noncrossing diagonals of a convex polygon into polygons whose number of
sides is congruent to $b$ modulo $a$. For the number of such partitions made by
a fixed number of diagonals, we give both a recurrence relation and an explicit
representation in terms of partial Bell polynomials. We use basic properties of
these polynomials to efficiently incorporate restrictions on the type of
polygons allowed in the partitions.
