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Math Help - Number Theory

  1. #1
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    Lightbulb Number Theory

    An unrestricted partition (order does not count, equality of sizes is ok) partition is called self-conjugate if it is identical with its conjugate. E. g 8 = 4 + 2 + 1 + 1.

    Show that the number of self conjugate unrestricted partitions of n is equal to the number of partitions of n into distinct odd parts

    (optional) express this result as an identity of generating functions.
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  2. #2
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    Quote Originally Posted by Waikato View Post
    An unrestricted partition (order does not count, equality of sizes is ok) partition is called self-conjugate if it is identical with its conjugate. E. g 8 = 4 + 2 + 1 + 1.

    Show that the number of self conjugate unrestricted partitions of n is equal to the number of partitions of n into distinct odd parts

    (optional) express this result as an identity of generating functions.
    A partition \lambda is self-conjugate if \lambda=\lambda' in terms of Ferrers diagram.

    The bijection can be shown graphically. See wiki.

    In your example, (4,2,1,1) corresponds to 7+1 in the link.

    The generating function for this is \prod_{\text{k=odd}}(1+x^k)=(1+x)(1+x^3)(1+x^5) \cdots.
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  3. #3
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    Wink partition

    Hello,

    Have you shown that the number of self conjugate unrestricted partitions of n is equal to the number of partitions of n into distinct odd parts?

    Thanks for your answers
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  4. #4
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    Quote Originally Posted by Waikato View Post
    Hello,

    Have you shown that the number of self conjugate unrestricted partitions of n is equal to the number of partitions of n into distinct odd parts?

    Thanks for your answers
    Yes. If your definition of an "unrestricted" partition is simply a partition without any further constraint being given.

    See the claim in the above link:

    Claim: The number of self-conjugate partitions is the same as the number of partitions with distinct odd parts.

    For instance, (5, 5, 4, 3, 2) |- n, where n=19, is a self-conjugate partition. It converts into the 9+7+3=19. Read the link I provided. It will give you some explanations graphically.

    The link I provided only shows the sketch of the proof. I encourage you to make a full proof on your own by using the sketch of the proof in the link.
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