Stirling numbers - hard proofs

[Meekah]

I have problem with prooving those two identities. Any help would be much appriciated!

Show that:
a)\(\displaystyle \begin{Bmatrix}

m+n+1\\ m

\end{Bmatrix}

= \sum_{k=0}^{m} k \begin{Bmatrix}

n+k\\k

\end{Bmatrix}\)

b)\(\displaystyle \sum_{k=0}^{n} \begin{pmatrix}

n\\k

\end{pmatrix}

\begin{Bmatrix}

k\\m

\end{Bmatrix}

= \begin{Bmatrix}

n+1\\m+1

\end{Bmatrix}\)

Where:
\(\displaystyle \begin{Bmatrix}

k\\m

\end{Bmatrix}\)

is a Stirling number of the second kind.

 

[Deleted member 4993]

I have problem with prooving those two identities. Any help would be much appriciated!

Show that:
a)\(\displaystyle \begin{Bmatrix}

m+n+1\\ m

\end{Bmatrix}

= \sum_{k=0}^{m} k \begin{Bmatrix}

n+k\\k

\end{Bmatrix}\)

b)\(\displaystyle \sum_{k=0}^{n} \begin{pmatrix}

n\\k

\end{pmatrix}

\begin{Bmatrix}

k\\m

\end{Bmatrix}

= \begin{Bmatrix}

n+1\\m+1

\end{Bmatrix}\)

Where:
\(\displaystyle \begin{Bmatrix}

k\\m

\end{Bmatrix}\)

is a Stirling number of the second kind.

You are putting up problems - without showing any work.

Please tell us the definition of Sterling numbers - and their properties relevant to the problem.

Please share your work with us, indicating exactly where you are stuck - so that we may know where to begin to help you.