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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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    @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

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

Etc.

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    Well, I'm currently reading that..starting to get the hang of it.. – 2012-12-02