A fountain is an arrangement of coins where each coin above the bottom row rests on two coins on the row below. The numbers of coin fountains with a base of size —that is, with coins in the bottom row—are the Catalan numbers . This can be seen by mapping a given coin fountain to a corresponding lattice path.

Snapshots 1, 2: For a given base size , the minimum number of coins in a fountain is —no coins above the base—and the maximum number is the triangular number . Between these extremes, there are multiple fountains with a base size and between and coins. (There is only one fountain with base and coins: a triangle missing its top.)

Snapshot 3: Each coin fountain corresponds to a lattice path, and this mapping illustrates the equivalence between coin fountains and Catalan numbers.

Reference

[1] R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge: Cambridge University Press, 1999 p. 228.