Coin Fountains with a Given Base
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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.
Contributed by: Robert Dickau (January 2012)
Open content licensed under CC BY-NC-SA
Snapshots
Details
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.
Permanent Citation