Chapter 8, Problem 12E

(a)

To determine

To prove: The Stirling numbers of the second kind $S\left(n,1\right)=1,\left(n\ge 1\right)$.

(b)

To determine

To prove: The Stirling numbers of the second kind $S\left(n,2\right)=2^{n-1}-1,\left(n\ge 2\right)$.

(c)

To determine

To prove: The Stirling numbers of the second kind $S\left(n,n-1\right)=\left(\begin{array}{l}n\\ 2\end{array}\right),\left(n\ge 1\right)$.

(d)

To determine

To prove: The Stirling numbers of the second kind $S\left(n,n-2\right)\text{=}\left(\begin{array}{l}n\\ 3\end{array}\right)\text{+3}\left(\begin{array}{l}n\\ 4\end{array}\right)\text{, }\left(n\ge 2\right)$.