What does an expression $[x^n](1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}(1-x^4)^{-1}...$ mean? I came across the function that describes number of partitions of $n$ (I mean partitions like $5=4+1=3+2=3+1+1$ and so on.
There was defined a Cartesian product: $$\{0,1,1+1,1+1+1,...\}\times\{0,2,2+2,2+2+2,...\}\times\{0,3,3+3,...\}\times...$$ so that each partition is an element of finite sum (sum over components is finite) of this product. For example $1+2+2=(1,2+2,0,0,...)$.
And then followed something I didn't understand: author introduced polynomials in the following way:
$$\begin{align}1:1+x+x^2+x^3+...&=(1-x)^{-1}\\2:1+x^2+x^4+x^6+...&=(1-x^2)^{-1}\\3:1+x^3+x^6+x^9+...&=(1-x^3)^{-1}\end{align}\\...$$
(powers are corresponding elements of product), then multiplied all the sums and stated, that number of partitions of $n$ is:
$$[x^n](1-x)^{-1}(1-x^2)^{-1}(1-x^3)^{-1}...$$
Later he showed an example, but it was not numeric – $x$ was left in the expression.
So how do I understand this?
Thanks in advance!

Often the notation
$$
[x^n](a_0+a_1x+a_2x^2+a_3x^3+\dots+a_nx^n+\dots)
$$
is used to mean the coefficient of $x^n$ in the given polynomial or power series ($a_n$ in the series I've given).
The formulas presented by the author are not polynomials, but power series. Polynomials are finite versions of power series. That is,
$$
\frac1{1-x}=1+x+x^2+x^3+x^4+\dots
$$
never terminates; there is no highest power of $x$.
For example,
$$
\left[x^6\right]\frac1{1-x}\frac1{1-x^2}\frac1{1-x^3}\frac1{1-x^4}\cdots=11
$$
