1)I was asked to find a partition function , where each number appears only once.
for example, for n=2 - 1+1 is not good but 2 is.
I think the function is : $\prod\limits_{k=1}^{\infty}(1+q)^k$,what do you think?
2)I was asked to find a partition function , where only odd numbers can appear.
for example, for n=2 - 1+1 is good but 2 is not.
I think the function is :
$\prod\limits_{k=1}^{\infty}1/(1-q^{2k-1}) $,what do you think?
3)I was asked to show that the partition function , where each number appears only once equals to the partition function , where only odd numbers can appear.
I tried to begin like this : $\prod\limits_{k=1}^{\infty}1/(1-q^{2k-1})=\prod\limits_{k=1}^{\infty}(1+q)^k $
and I got to: $\prod\limits_{k=1}^{\infty}(q^k (q^{k-1}+q^{2k-1}-1)/(1-q^{2k-1}) $ but now I am stuck