Explain this generating function I have a task:
Explain equation:
$$\prod_{n=1}^{\infty}(1+x^nz) = 1 + \sum_{n=m=1}^{\infty}\lambda(n,m)x^nz^m  $$
$\lambda(n,m)$ - is number of breakdown $n$ to $m$ different numbers (>0) 
It's hard to me.
The only thing that I can explain is:
$\prod_{n=1}^{\infty}(1+x^nz) = (1+xz)(1+x^2z)(1+x^3z)... $
It has to be any generating function. For example for string:
$(1, z, 0,0,0,0,...)$
$(1,0,z,0,0,0,...) $
$(1,0,0,z,0,0,....)$
$$...$$
So far I have use only generating function and I have one variable $x$. Here I have two variables: $x$ and $z$. I don't understand this. This task is very difficult to me.
Could somebody help me ?
 A: Multiplying formal power series in multiple indeterminates is not really much more complicated than doing so for univariate ones. The main difference is that each term has not a single degree, but separate degrees for each of the $k$ indeterminates, so the coefficient is attached to a point of $\def\N{\Bbb N}\N^k$ rather than of$~\N$.
With an infinite product like this one, the first important point is that in the expansion, only those products of terms of the individual factors contribute that are well defined (have finite degrees), which amounts to saying one is allowed to choose a term distinct from $1$ only finitely many times. So here one can choose for finitely many different values of $n$ the term $x^nz$, and the term $1$ for all other values of $n$. The product of these terms is $x^sz^m$, where $s$ is the sum of the values $n$ selected, and $m$ is their number. Each such term is contributed with coefficient$~1$, so it suffices to count the number of times a given term $x^sz^m$ is contributed. This is precisely the number of $m$-element subsets of $\N_{>0}$ with sum$~s$. Ordering the elements of such a subset in decreasing order, one obtains a partition of $s$ into $n$ distinct nonzero parts (also called a strict partition of $s$ with $n$ parts), which is apparently the interpretation you are after.
Since parts are nonzero, $s$ can only be zero if $m$ is also, and the converse is obvious, so if one likes $\sum_{s,m\in\N}\lambda(s,m)x^sz^m$ can be rewritten as $1+\sum_{s,m\in\N_{>0}}\lambda(s,m)x^sz^m$ (it doesn't buy much though).
