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Thread: Partition of a number with no 1s

  1. #1
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    Partition of a number with no 1s

    Hi.

    I have problems solving this problem:

    How many partitions of a number n are there, if none of the components of the partition equals 1?

    What I've got so far is that for
    n=1 we have 0,
    n=2 and n=3 we have 1
    n=4 and n=5 we have 2
    n=6 and n=7 we have 3

    but then for n=8 we get 7 and for n=9 we get 8.

    Unless I'm making mistakes while counting. Anyway, I do not see any relations here.

    Could someone help me?
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  2. #2
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    Re: Partition of a number with no 1s

    Quote Originally Posted by wilhelm View Post
    How many partitions of a number n are there, if none of the components of the partition equals 1?
    What I've got so far is that for
    n=1 we have 0,
    n=2 and n=3 we have 1
    n=4 and n=5 we have 2
    n=6 and n=7 we have 3
    but then for n=8 we get 7 and for n=9 we get 8.
    Unless I'm making mistakes while counting. Anyway, I do not see any relations here.
    This is a rather deep question.
    If you look at this webpage, you will see how deep.

    The function you want is $\displaystyle p(2,n)$. But there is no easy way to define that function.
    Thanks from wilhelm
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  3. #3
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    Re: Partition of a number with no 1s

    I'm sorry, I don't see that; $\displaystyle p(2,n)$ describes the number of n using 2 numbers, right? But in the problem above

    $\displaystyle 7=7+0=5+2=4+3=2+2+3$ but not $\displaystyle 6+1,5+1+1,4+1+1+1,....$

    $\displaystyle 8=8+0=6+2=5+3=4+4=2+3+3=2+2+4=2+2+2+2$ but not for example $\displaystyle 7+1, 6+1+1, 5+1+1+1, ...$

    That's why I don't understand why we should consider the function $\displaystyle p(2,n)$
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  4. #4
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    Re: Partition of a number with no 1s

    Quote Originally Posted by wilhelm View Post
    I'm sorry, I don't see that; $\displaystyle p(2,n)$ describes the number of n using 2 numbers, right? But in the problem above
    Here is the quote:
    For any given value of k, partitions counted by p(k, n) fit into exactly one of the following categories:
    smallest addend is k
    smallest addend is strictly greater than k.
    Thanks from wilhelm
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