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Our professor gave us the theorem:

$S(n+1,m+1)=\sum\limits_{k=m}^n \binom{n}{k} S(k,m)$,

where $S(n,m)$ denotes the number of partitionings of a set with $n$ elements into $m$ blocks.

I don't want a proof of this equation as Wikipedia and our professor gave it to us. I just don't understand the intuition behind it.

Could someone explain using for example gifts and packages ?

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    Do you know what is a partition of a set with $n$ elements? The Stirling numbers of the second kind, denoted by $S(n,k)$, denoted the number of different ways to make a partition of a set with $n$ elements into $m$ subsets. – 2017-02-28
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    I think so yes, its just splitting all elements of the set in groups. – 2017-02-28
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    exactly, but the correct name is not "groups", instead we use the term subset. – 2017-02-28

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The real key is that $\binom{n}{k}=\binom{n}{n-k}$. So you take $n+1$ gifts labeled $\{1,\dots n+1\}.$ Take gift $n+1$ and $n-k$ other gifts, and put them in the first package, then partition the remaining $k$ gifts into $m$ packages.

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    ok i kind of get it, but why start the sum from m ? – 2017-02-28
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    Technically, you could start the sum before $m$, but $S(k,m)=0$ if $k$k$ gifts into $m$ packages without some packages being empty, if $k – 2017-02-28
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Let show an example so you can visualize this. Let the set defined as $A:=\{a,b,c\}$, then the set $A$ have $3$ elements. The number of ways to make a partition of $A$ in two subsets is $3$, that is

$$\{\{a,b\},\{c\}\},\quad\{\{a,c\},\{b\}\},\quad\text{and}\quad\{\{b,c\},\{a\}\}$$

then $S(3,2)=3$.