Description:Catalan numbers Cn form one of the most frequently used sequences of natural numbers. They appear as answers to numerous enumerative problems, completely different from the first glance.As an example of their appearance one can take the number of expressions containing n pairs of parentheses which are correctly matched. For example, C3=5 as you can see from the following list: ()()(),---- ()(()),---- (())(),---- (()()),---- ((())) (why is this list complete?) Catalan numbers not only offer a unified answer to a wide selection of combinatorial problems, they also may serve as a good starting point if you want to learn combinatorics of polytopes or cluster algebras. PrerequisitesThere are no formal prerequisites, but you will need to see similarities between objects of different nature, e.g. from algebra, discrete mathematics and probability.ResourcesThe wikipedia page for Catalan numbers contains many examples of problems where these numbers arise, it also contains many external references. In particular, the Catalan addendum by Richard Stanley contains dozens of equivalent definitions. The following elementary book includes Catalan numbers in a series of other types of distinguished number sequences:
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email: Pavel Tumarkin