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    #1
    Find the number of different combinations of $ \displaystyle r$ natural.numbers that add upto $ \displaystyle n$

    I tried this for quite a fair amount.of.time but.couldn't figure it out.
    Last edited by mathworker; July 27th, 2013 at 18:45.

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    #2
    Quote Originally Posted by mathworker View Post
    Find the number of different combinations of $ \displaystyle r$ natural.numbers that add upto $ \displaystyle n$

    I tried this for quite a fair amount.of.time but.couldn't figure it out.
    This is known as the problem of Partitions of an Integer.
    .

    If you want print material see Ivan Niven's Mathematics of Choice, chapter six.

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    #3 Thread Author
    Thanks for the link,I have gone through it.
    But as far as I understood partition function doesn't give the number of partitions of specific cardinality.I mean if we want only the partitions that contains $ \displaystyle r$ terms for example or can we define a restricted partition function that can do the job?If we can define how can we approximate such restricted $ \displaystyle p(x)$

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    #4
    Quote Originally Posted by mathworker View Post
    Thanks for the link,I have gone through it.
    But as far as I understood partition function doesn't give the number of partitions of specific cardinality.I mean if we want only the partitions that contains $ \displaystyle r$ terms for example or can we define a restricted partition function that can do the job?If we can define how can we approximate such restricted $ \displaystyle p(x)$
    Well I did say that the webpage is only fair. I dislike its notation.
    I suggest that you try to find Niven's book.

    Example: The number of partitions of 6 into 3 summands is three:
    $ \displaystyle \begin{align*} 6 &= 1+1+4\\ &=1+2+3\\ &=2+2+2\end{align*}$

    That is $ \displaystyle p_3(6)-p_2(6).$

    There is a clear recursive definition of $ \displaystyle p_k(n). $

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    #5
    Quote Originally Posted by mathworker View Post
    Find the number of different combinations of $ \displaystyle r$ natural.numbers that add upto $ \displaystyle n$
    There is about this. Of course, the problem is tricky if "combination" is used in its technical sense to mean a set. In contrast, permutations (i.e., ordered sequences) of summands are called compositions (rather than partitions). Their number is .

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