Denote the partition function by $p_k(n)$, and define it as a count of the number of possible sequences of positive integers $a+b+c+...=n$ where the $a,b,c,...$ are not necessarily distinct (so that, for example, $1+3+4$ is not counted as distinct from $1+4+3$).
I know Wikipedia can be unreliable. But I read here that "The multiplicative inverse of its [the partition function's] generating function is the Euler function." Does this mean Euler's totient function $\phi(n)$? It's impossible to tell from the context or the links. I've searched around and I have come out none the wiser.
The Wikipedia link asserts that the "Euler function" is given by
But it goes into no further detail apart from cryptic references to "$q$-series. And when you follow links to that, you end up with no answers at all.
Could someone please explain what the quote above means - ideally in algebraic form? I understand what a generating function is (though the article doesn't say which sort it is referring to), but I would really like to see that this under-explained statement as a formula.