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Math Help - What are the following types of numbers called?

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    What are the following types of numbers called?

    Let n be a natural number, then s is the number of ways to write n as the sum of at least one natural number. What is the number s called?
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    Quote Originally Posted by Chris11 View Post
    Let n be a natural number, then s is the number of ways to write n as the sum of at least one natural number. What is the number s called?
    If the addends are not necessarily distinct and order is not considered, then we have Partition Function P. And here is the Wikipedia article.
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    What about when order is considered?
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    Quote Originally Posted by Chris11 View Post
    What about when order is considered?
    This is the number of compositions of n.
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    Thanks for replying. I made the conjecture that the number of compositions is  2^{n-1}, but I haven't been able to prove it. I don't understand the proof on wikipedia much either... I shamefully don't know anything about combinatorical proof..
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    Quote Originally Posted by Chris11 View Post
    Thanks for replying. I made the conjecture that the number of compositions is  2^{n-1}, but I haven't been able to prove it. I don't understand the proof on wikipedia much either... I shamefully don't know anything about combinatorical proof..
    The proof is not too hard, let's try an example: n=3.

    1+1+1 = (3)
    1+1,1 = (2,1)
    1,1+1 = (1,2)
    1,1,1 = (1,1,1)

    You can look at the "+" and "," symbols as 0 and 1.

    00
    01
    10
    11

    Does it make more sense now?
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    yeah, it makes alot more sense. Thanks.
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