I'm trying to prove the following Integer Partition claim :
$p$(n│even number of ODD parts) = $p$(n│distinct parts ,number of ODD parts is even) .
So I tried to prove a stronger claim :
$p$(n│number of odd parts) = $p$(n│number of distinct parts) :
Using generating functions :
DISTINCT PARTS = $$(1+x)\cdot(1+x^2 )\cdot(1+x^3 )\cdot(1+x^4 )\cdot\cdot\cdot\cdot$$
= $$\frac {1-x^2} {1-x} \cdot \frac {1-x^4} {1-x^2} \cdot \frac {1-x^6} {1-x^3} \cdot \frac {1-x^8} {1-x^4}\cdot\cdot\cdot\cdot$$
$$ \frac {1} {1-x} \cdot \frac {1} {1-x^3} \cdot \frac {1} {1-x^5} \cdot \frac {1} {1-x^7} \cdot\cdot\cdot\cdot $$
= ODD PARTS
But I'm not sure regarding the proof , does it really prove that even number of odd parts equals distinct parts ,where number of ODD parts is even ?
Thanks