Generating function for partitions into parts

Question

My question is short and (maybe) simple: What is the generating function for partitions into distinct parts equal to $2, 5$ or $7$?

My idea is to use Euler's theorem: $$\sum p(k)x^k=\prod\frac{1}{1-x^k}.$$

But how is his theorem concretely applicable to my problem?

Answer 1

It is $$(1+x^2)(1+x^5)(1+x^7)$$

Answer 2

In general, the formula is as follows. Let $S$ be the set of allowed part sizes, and for each $n\in S$, let $T_n$ be the set of allowed multiplicities of a part of of size $n$. Then the number of partitions with these allowed part sizes and multiplicities has the ordinary generating function $$ \prod_{n\in S}\left(\sum_{k\in T_n} x^{nk}\right). $$ In your case, $S=\{2,5,7\}$ and $T_2=T_5=T_7=\{0,1\}$.

When you want all distinct parts, each $T_n=\{0,1\}$, so each factor is $1+x^n$. When you allow all part multiplicities, each $T_n=\mathbb{N}\cup\{0\}$, so each factor is $$ \sum_{k=0}^{\infty} x^{nk}=\frac{1}{1-x^n}. $$