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Thread: Generating Functions and Integer Partitions

  1. #1
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    Generating Functions and Integer Partitions

    A.Let A(n) be the number of partitions n in which no part occurs more than twice. Find the generating function.

    My Answer:
    $\displaystyle (1 +x)(x+x^{2})(x^{2}+x^{3})...........(x^{i}+x^{i+1} )$
    $\displaystyle \prod_{i=0}^{\infty}$$\displaystyle (x^{i}+x^{1+i})$


    B. let B(n) be the number of partitions of n in which no part is a multiple of 3. Find the Generating function.

    $\displaystyle \dfrac{1}{1-x}$-$\displaystyle \dfrac{1}{1-x^{3}}$

    I'm still new with these, I'm sure I messed up, can anyone help me out?
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  2. #2
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    I've changed my answers, but help need to decide if they're right

    A)$\displaystyle \prod_{i=1}^{\infty}(1+x^{i}+x^{2i})$

    B)$\displaystyle \prod_{i=1}^{\infty}\dfrac{(1-x^{3i})}{1-x^{i}}$
    Last edited by Jrb599; Apr 9th 2008 at 08:26 AM.
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  3. #3
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    Quote Originally Posted by Jrb599 View Post
    A)$\displaystyle \prod_{i=1}^{\infty}(1+x^{i}+x^{2i})$
    $\displaystyle (1+x+x^2)(1+x^2+x^4)(1+x^3+x^6)...$
    What are the different way of getting $\displaystyle x^4$?
    $\displaystyle x^0x^4+x^1x^3+x^2x^2=x^4$.

    This cannot be it because $\displaystyle 1+3,4$ are the only such summations.
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  4. #4
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    Quote Originally Posted by ThePerfectHacker View Post
    $\displaystyle (1+x+x^2)(1+x^2+x^4)(1+x^3+x^6)...$
    What are the different way of getting $\displaystyle x^4$?
    $\displaystyle x^0x^4+x^1x^3+x^2x^2=x^4$.

    This cannot be it because $\displaystyle 1+3,4$ are the only such summations.
    Don't forget 2+2

    2 can occur twice, just not more than twice.
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