Number of Paths on Graph along X-axis using Diagonal Steps

Theorem

The number of different paths that can be taken on a Cartesian plane from the origin to $\tuple {2 n + 2, 0}$, using only diagonal steps, and never touching the $x$-axis except at the beginning and the end of the path, is the Catalan number $C_n$:

Proof

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Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $42$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $42$