I am suppose to prove that the number of partitions of $n$ in which each part appears $2$, $3$, or $5$ times equals the number partitions of $n$ in to parts which are congruent to $2$, $3$, $6$, $9$, or $10$ modulo $12$.
I tried using Remmels Thm in order to prove this but I ran into a problem classifying the pairwise disjoint multisets. I now think the approach that should be used is find the generating function for each and show equality using algebra. I think the generating function for
$A$: partitions of $n$ in which each part appears $2$, $3$, or $5$ times
$$ G(A) = (1+x^2+x^4+...)(1+x^3+x^6+..)(1+x^5+x^10+...) = \frac{1}{1-x^2}\frac{1}{1-x^3}\frac{1}{1-x^5} $$
$B$: partitions of $n$ into parts which are congruent to $2$,$3$,$6$,$9$, or $10$ modulo $12$
$$ G(B) = \frac{1}{(1-x^{12k-2})(1-x^{12k-3})(1-x^{12k-6})(1-x^{12k-9})(1-x^{12k-10})} $$
Is this correct? Not sure how to manipulate the functions into each other