How many different clusterings are possible for certain number of clusters and elements

Question

let $C = \{C_1,... ,C_2 \}$ be a clustering of $k$ Clusters and I have a set $G$ with $n$ objects in total. Then I want to find different clusterings which meet the basic condition of a clustering:
$C_j \subset G$ and $C_i \cap C_j$ for $i \neq j$ and $G = \cup_{i=1}^{k} C_i$.

My script than states without proof that the number of possible clusterings C with k Clusters and n elements is given by

$$ s(k,n) = \frac{1}{k!}\sum_{j=1}^k (-1)^{k-j} \binom{k}{j}j^n $$

I couldn't find that formular by googling and I wanted to know how this formula is derived (at least heuristically), since the term $(-1)^{k-j}$ is really counter-intuitive for me.

Thank you in advance.

Answer 1

Hint: The numbers ${n\brace k}$ or $S(n,k)$ are called Stirling numbers of the Second kind and give the number of subdivisions of a set of $n$ objects into $k$ partitions.

The expression $${n\brace k}=\frac{1}{k!}\sum_{j=1}^k (-1)^{k-j} \binom{k}{j}j^n $$ is one explicit formula for the Stirling numbers of the second kind.

Notes:

• The factor $(-1)^{k-j}$ often indicates an application of the Inclusion-Exclusion principle.

• A short combinatorial answer is stated in this MSE post.

• The symbol $s(n,k)$ or $\left[n\atop k\right]$ is usually reserved for the Stirling numbers of the first kind.