Consider the problem of stacking coins in the plane such that the bottom row consists of $n$ consecutive coins.
Prove that the number of coin configurations satisfies the Catalan recurrence.
I understand I need some sort of bijection to Dyck paths, possibly with {$(1,1),(1.-1)$} steps in the plane, and then the length of the path would be the number of such configurations for every $n$. However thats not really a proof..