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One Interpretation of a Catalan Number Recurrence Relation

The recurrence relation

, with

, defines the Catalan numbers. One interpretation of the Catalan numbers, in turn, is the number of ways to tile a stairstep shape of height

using

rectangles.

To see that the two descriptions are equivalent, begin a stairstep tiling by placing a green rectangle of dimensions

. Fit tilings from a previous stage—that is, tiling for the smaller blue and purple stairstep shapes—into the remaining spaces.

(15 lines omitted)

Contributed by: Robert Dickau

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