Abstract
The Fibonacci numbers are pretty cool, but in modern algebraic combinatorics, the most interesting sequence is the sequence of Catalan numbers: $$\begin{aligned} 1, 1, 2, 5, 14, 42, 132, 429,\ldots \end{aligned}$$ The remarkable ability of these numbers to pop up in surprising locations has led some to joke that a combinatorics paper is not complete until the Catalan numbers have made an appearance. There are now more than two hundred distinct combinatorial interpretations for Catalan numbers!
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