We will learn about a combinatorics tool: Stirling Numbers of First Kind. Here is a video to get you started.
Problem 1
Show that
$$
s(r, n)=s(r-1, n-1)+(r-1) s(r-1, n)
$$
where $r, n \in \mathbf{N}$ with $n \leq r$
Problem 2
Show the following:
$$ s(r, 0)=0 \text { if } r \geq 1 $$
$$ s(r, r)=1 \text { if } r \geq 0 $$
$$ s(r, 1)=(r-1)! \text { for } r \geq 2 $$
$$ s(r, r-1)={{r} \choose {2}} $$
