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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). –  dtldarek Dec 2 '12 at 14:40
    
@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 :) –  VISHNU VIVEK Dec 3 '12 at 11:52
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up vote 4 down vote accepted

I'd read the wikipedia article in a different section order that 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. –  VISHNU VIVEK Dec 2 '12 at 14:53
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That's straight from the Wikipedia article -- will copy their example... –  Mark Hurd Dec 2 '12 at 14:55
    
Well, I'm currently reading that..starting to get the hang of it.. –  VISHNU VIVEK Dec 2 '12 at 14:59
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