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.