Generating Functions and Integer Partitions

**Jrb599** · April 8th 2008, 07:23 PM

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

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

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

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

I'm still new with these, I'm sure I messed up, can anyone help me out?

**Jrb599** · April 9th 2008, 06:41 AM

I've changed my answers, but help need to decide if they're right

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

B) $\prod_{i=1}^{\infty}\dfrac{(1-x^{3i})}{1-x^{i}}$

**ThePerfectHacker** · April 9th 2008, 07:56 AM

Originally Posted by Jrb599

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

$(1+x+x^2)(1+x^2+x^4)(1+x^3+x^6)...$
What are the different way of getting $x^4$ ?
$x^0x^4+x^1x^3+x^2x^2=x^4$ .

This cannot be it because $1+3,4$ are the only such summations.

**Jrb599** · April 9th 2008, 08:28 AM

Originally Posted by ThePerfectHacker

$(1+x+x^2)(1+x^2+x^4)(1+x^3+x^6)...$
What are the different way of getting $x^4$ ?
$x^0x^4+x^1x^3+x^2x^2=x^4$ .

This cannot be it because $1+3,4$ are the only such summations.

Don't forget 2+2

2 can occur twice, just not more than twice.

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