Generating function with Stirling's numbers of the second kind It's very easy to prove that:
$$\sum_k \left\{k\atop n\right\}z^k=\frac{z^n}{(1-z)(1-2z)...(1-nz)}$$
But this generating function looks very pretty, so my question is: does this identity have some simple combinatorial interpretation?
 A: Yes, this equation has a combinatorial interpretation. 
The number $\left\{k\atop n\right\}$ counts the number of partitions of the linearly ordered set $\mathbf k=\{1, \dots, k\}$ into $n$ nonempty subsets. Let us see how one can think of such a partition. Below, you can see a partition of $\mathbf{15}$ into $4$ nonempty subsets (the linear order goes from left to right).

The data of this partition is equivalent to the following data:

Here, a vertical line was inserted just before a new color is encountered when going from left to right. The colors were ordered by their order of occurence:
$$\text{Red}<\text{Blue}<\text{Green}<\text{Pink}$$
and were assigned the numbers $0,1,2,3$ respectively. Now, remark that the data of the above decomposition is also equivalent to the data of the diagram 

because the obscured numbers are just $0,1,2,3$ in order of occurence. The initial partition can be completely reconstructed from this last diagram. I'll leave it up to you to check that this decomposition yields the claimed identity of power series.
