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I own a copy of Richard P. Stanley's book on Catalan numbers. He lists 214 combinatorial interpretations of Catalan numbers (inviting readers to become experts by finding all the 214 * 213 = 45582 bijections)...

At some point (in 2004, my God) I was interested in the generalization from squares to rectangles (learning from Richard K. Guy's papers): http://www.meyer-idstein.de/article_walks.pdf

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Stanley is awesome.

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