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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

The number of legal ways to arrange parentheses is exactly the number of chords that do not intersect!

**In mathematics, you don't understand things. You just get used to them.**

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I somewhat got it now.

What does it mean when the expression is a nested one like (()) ?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

Nothing at all except it is a legal ordering of the parentheses which (()( is not.

**In mathematics, you don't understand things. You just get used to them.**

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One more thing I would like to ask is how can you make sure that the cases of legal expression and nonintersecting chords are equivalent?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

That is done by empirically getting 1,2,5,14 and then consulting the Oracle at Delphi. This is one of the chief advantages of doing experimental - computational mathematics!

**In mathematics, you don't understand things. You just get used to them.**

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Oh so you just compare the cases?

What is the oracle of Delphi?

*Last edited by Agnishom (2013-04-04 02:23:30)*

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

The Oracle at Delphi is from Greek mythology. Apollo would speak to anyone consulting the Oracle providing unlimited knowledge and understanding of all things.

**In mathematics, you don't understand things. You just get used to them.**

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Wow!

So, its the 8th Catalan number/7*5*3

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

4th Catalan number. 4 for the number of chords over 7 * 5 * 3 * 1 = 2 / 15

**In mathematics, you don't understand things. You just get used to them.**

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Thanks a lot but please answer the first question in #56

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

Compare the cases? I am not following you.

**In mathematics, you don't understand things. You just get used to them.**

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I mean is it just by mere observation that you know that the cases of the chords and that of the brackets are equivalent?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

The first few numbers you do by hand or you use a computer. Then you take them over to the Oracle( remember ) and it will do the rest.

**In mathematics, you don't understand things. You just get used to them.**

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But you need a proof to say that that will work for all of them!

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

That proof is provided by the site I am referring to. This is a well known combinatorics problem solved long ago.

But you need a proof to say that that will work for all of them!

We will discuss this somewhat old fashioned concept a little bit later. Just let me say that much math is done with a proof connected to it.

**In mathematics, you don't understand things. You just get used to them.**

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Experimental math? Huh?

But how did you know that those two series are the same?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

Once we have some portion of the sequence we can now look it up and find all pertinent information.

**In mathematics, you don't understand things. You just get used to them.**

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Maybe I would look up the Online Enclycopedia of integer sequences

What would you do if there was no internet?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

I would use the book. The internet is also one of the cornerstones of experimental math.

Maybe I would look up the Online Enclycopedia of integer sequences

Wiser than the Oracle at Delphi it is! Check it and you will see your whole problem already solved. Recurrences, generating functions, references.

Unfortunately, I did not use it and had to rediscover America.

**In mathematics, you don't understand things. You just get used to them.**

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If you had nothing, how could you do it all from a scratch?

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

Do I need to? I also could not get 1000! from nothing. Technology is here, it is ruining the environment, we at least should use it.

**In mathematics, you don't understand things. You just get used to them.**

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You do not need 1000! at all.

This problem is simpler

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

Once we take the formula for the Catalan numbers and put it over the formula for the double factorial and simplify we will get the formula that I posted earlier.

Now if you can follow that we can talk a bit about your statement of needing a proof in the ancient greek sense of a proof.

**In mathematics, you don't understand things. You just get used to them.**

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I somewhat understand

'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'

'You have made another human being happy. There is no greater accomplishment.' -bobbym

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 90,615

Did you know that many famous mathematicians were not very big on proofs. The names of Eudoxus, Poincare and Bernhard Riemann come to mind.Did you know your countryman, who many call the greatest mathematician of the twentieth century never proved anything?

**In mathematics, you don't understand things. You just get used to them.**

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