# Sum over k of Stirling Numbers of the Second Kind of k+1 with m+1 by n choose k by -1^k-m

## Theorem

Let $m, n \in \Z_{\ge 0}$.

$\displaystyle \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom n k \left({-1}\right)^{n - k} = \left\{ {n \atop m}\right\}$

where:

$\displaystyle \left\{ {k + 1 \atop m + 1}\right\}$ etc. denotes a Stirling number of the second kind
$\dbinom n k$ denotes a binomial coefficient.

## Proof

The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $P \left({n}\right)$ be the proposition:

$\displaystyle \forall m \in \Z_{\ge 0}: \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom n k \left({-1}\right)^{n - k} = \left\{ {n \atop m}\right\}$

### Basis for the Induction

$P \left({0}\right)$ is the case:

 $\displaystyle$  $\displaystyle \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom 0 k \left({-1}\right)^{- k}$ $\displaystyle$ $=$ $\displaystyle \sum_k \left\{ {k + 1 \atop m + 1}\right\} \delta_{0 k} \left({-1}\right)^{- k}$ Zero Choose n $\displaystyle$ $=$ $\displaystyle \left\{ {1 \atop m + 1}\right\}$ all terms vanish except for $k = 0$ $\displaystyle$ $=$ $\displaystyle \delta_{1 \left({m + 1}\right)}$ Stirling Number of the Second Kind of 1 $\displaystyle$ $=$ $\displaystyle \delta_{0 m}$ Definition of Kronecker Delta $\displaystyle$ $=$ $\displaystyle \left\{ {0 \atop m}\right\}$ Stirling Number of the Second Kind of 0

So $P \left({0}\right)$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

Now it needs to be shown that, if $P \left({r}\right)$ is true, where $r \ge 1$, then it logically follows that $P \left({r + 1}\right)$ is true.

So this is the induction hypothesis:

$\displaystyle \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom r k \left({-1}\right)^{r - k} = \left\{ {r \atop m}\right\}$

from which it is to be shown that:

$\displaystyle \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom {r + 1} k \left({-1}\right)^{r + 1 - k} = \left\{ {r + 1 \atop m}\right\}$

### Induction Step

This is the induction step:

 $\displaystyle$  $\displaystyle \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom {r + 1} k \left({-1}\right)^{r + 1 - k}$ $\displaystyle$ $=$ $\displaystyle \sum_k \left\{ {k + 1 \atop m + 1}\right\} \left({\binom r k + \binom r {k - 1} }\right) \left({-1}\right)^{r + 1 - k}$ Pascal's Rule $\displaystyle$ $=$ $\displaystyle -\sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom r k \left({-1}\right)^{r - k} + \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom r {k - 1} \left({-1}\right)^{r - \left({k - 1}\right)}$ $\displaystyle$ $=$ $\displaystyle -\left\{ {r \atop m}\right\} + \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom r {k - 1} \left({-1}\right)^{r - \left({k - 1}\right)}$ Induction Hypothesis $\displaystyle$ $=$ $\displaystyle -\left\{ {r \atop m}\right\} + \sum_k \left({\left({m + 1}\right) \left\{ {k \atop m + 1}\right\} + \left\{ {k \atop m }\right\} }\right) \binom r {k - 1} \left({-1}\right)^{r - \left({k - 1}\right)}$ Definition of Stirling Numbers of the Second Kind $\displaystyle$ $=$ $\displaystyle -\left\{ {r \atop m}\right\} + \left({m + 1}\right) \sum_k \left\{ {k \atop m + 1}\right\} \binom r {k - 1} \left({-1}\right)^{r - \left({k - 1}\right)}$ $\displaystyle$  $\displaystyle + \sum_k \left\{ {k \atop m }\right\} \binom r {k - 1} \left({-1}\right)^{r - \left({k - 1}\right)}$ $\displaystyle$ $=$ $\displaystyle -\left\{ {r \atop m}\right\} + \left({m + 1}\right) \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom r k \left({-1}\right)^{r - k} + \sum_k \left\{ {k + 1 \atop m }\right\} \binom r k \left({-1}\right)^{r - k}$ Translation of Index Variable of Summation $\displaystyle$ $=$ $\displaystyle -\left\{ {r \atop m}\right\} + \left({m + 1}\right) \left\{ {r \atop m}\right\} + \left\{ {r \atop m - 1}\right\}$ Induction Hypothesis $\displaystyle$ $=$ $\displaystyle m \left\{ {r \atop m}\right\} + \left\{ {r \atop m - 1}\right\}$ $\displaystyle$ $=$ $\displaystyle \left\{ {r + 1 \atop m}\right\}$ Definition of Stirling Numbers of the Second Kind

So $P \left({r}\right) \implies P \left({r + 1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:

$\displaystyle \forall m, n \in \Z_{\ge 0}: \sum_k \left\{ {k + 1 \atop m + 1}\right\} \binom n k \left({-1}\right)^{n - k} = \left\{ {n \atop m}\right\}$

$\blacksquare$