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# What is Stirling Number of First Kind

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:

$$s(r, 0)=0 \text { if } r \geq 1$$

$$s(r, r)=1 \text { if } r \geq 0$$

$$s(r, 1)=(r-1)! \text { for } r \geq 2$$

$$s(r, r-1)={{r} \choose {2}}$$

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:

$$s(r, 0)=0 \text { if } r \geq 1$$

$$s(r, r)=1 \text { if } r \geq 0$$

$$s(r, 1)=(r-1)! \text { for } r \geq 2$$

$$s(r, r-1)={{r} \choose {2}}$$

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