# Generating Functions and Integer Partitions

• Apr 8th 2008, 07:23 PM
Jrb599
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.

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

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?
• Apr 9th 2008, 06:41 AM
Jrb599
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) $\prod_{i=1}^{\infty}\dfrac{(1-x^{3i})}{1-x^{i}}$
• Apr 9th 2008, 07:56 AM
ThePerfectHacker
Quote:

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.
• Apr 9th 2008, 08:28 AM
Jrb599
Quote:

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.