One Interpretation of a Catalan Number Recurrence Relation

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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.

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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.

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Contributed by: Robert Dickau (March 2011)
Open content licensed under CC BY-NC-SA


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