Thread: Generating Functions and Integer Partitions

1. 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.

$\displaystyle (1 +x)(x+x^{2})(x^{2}+x^{3})...........(x^{i}+x^{i+1} )$
$\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?

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

3. Originally Posted by Jrb599
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.

4. Originally Posted by ThePerfectHacker
$\displaystyle (1+x+x^2)(1+x^2+x^4)(1+x^3+x^6)...$
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.