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I have read the Wikipedia article for Catalan number and a number of other websites, but still couldn't understand it. Please explain it in simple terms, or using some examples. Thanks in advance!

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    Which part of Wikipedia's section "Applications in combinatorics" is not clear to you? As you can see, Catalan numbers have multiple interpretations (each of these is a fine example).2012-12-02
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    @dtldarek actually I thought if I couldn't understand the basic concept of catalan numbers, then what is the use in reading the applications.. then after Mark suggested, I read it and finally understood it.. Thanks :)2012-12-03

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I'd read the Wikipedia article in a different section order than it is currently written:

Applications in combinatorics

The $n$th Catalan number counts the number of different ways $n$ pairs of brackets can be correctly matched.

E.g. for $n = 3$ there are these distinct correctly matched pairs of brackets: $$((()))\space\space()(())\space\space()()()\space\space(())()\space\space(()())$$ (Etc.)

Properties

$C_0 = 1 \text{ and } C_{n+1} = \sum_{i=0}^nC_iC_{n−i} \text{ for } n\ge0$

Etc.

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    Hi Mark! you've mentioned "The nth Catalan number counts the number of different ways n pairs of brackets can be correctly matched". This line seems to make sense. Can you provide an example.2012-12-02
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    That's straight from the Wikipedia article -- will copy their example...2012-12-02
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    Well, I'm currently reading that..starting to get the hang of it..2012-12-02