Say I have a generating function $\Phi_\mathcal{A}$ for the set of partitions $\mathcal{A}$ which have no parts congruent to 2 mod 4, and I have the generating function for $\Phi_\mathcal{B}$ for the set of partitions $\mathcal{B}$ in which the parts divisible by 4 occur at most once, can I multiply the 2 generating functions together to obtain the generating function for the intersection of $\mathcal{A}$ and $\mathcal{B}$ (i.e for the partitions that satisfy both requirements)? And more importantly, is this a technique that I can use for arbitrary such conditions?
The reason I ask is because my homework assignment contains several questions which require me to give the generating function for partitions that have some property AND some other property, and I wanted to know if this technique was a good/valid one to use for these problems.
Also, is there a similar technique for giving the generating function for integer partitions that have some property OR some other property?
Thanks!
Multiplying them won’t do the job: that gives you the convolution of the corresponding sequences. (Actually, you can see that multiplication isn’t right by considering what happens when $\mathcal A=\mathcal B$: then you’d have to have $(\Phi_{\mathcal A})^2=\Phi_{\mathcal A}\Phi_{\mathcal B}=\Phi_{\mathcal A\cap\mathcal B}=\Phi_{\mathcal A}$, which you can’t reasonably expect.
In this case notice that the conditions allow unrestricted odd parts and restrict the even parts to multiples of $4$ occurring once, giving you
$$\prod_{k\ge 1}\frac1{1-x^{2k-1}}\cdot\prod_{k\ge 1}\left(1+x^{4k}\right)=\prod_{k\ge 1}\frac{1+x^{4k}}{1-x^{2k-1}}\;.$$