# Second Inversion Formula for Stirling Numbers

## Theorem

For all $m, n \in \Z_{\ge 0}$:

$\displaystyle \sum_k \left\{ {n \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{n - k} = \delta_{m n}$

where:

$\displaystyle \left\{ {n \atop k}\right\}$ denotes a Stirling number of the second kind
$\displaystyle \left[{k \atop m}\right]$ denotes an unsigned Stirling number of the first kind
$\delta_{m n}$ denotes the Kronecker delta.

## Proof

The proof proceeds by induction.

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

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

### Basis for the Induction

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

 $\displaystyle \sum_k \left\{ {0 \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{0 - k}$ $=$ $\displaystyle \sum_k \delta_{k 0} \left[{k \atop m}\right] \left({-1}\right)^{-k}$ Definition of Stirling Numbers of the Second Kind $\displaystyle$ $=$ $\displaystyle \left[{0 \atop m}\right]$ as all other terms vanish by $\delta_{k 0}$ $\displaystyle$ $=$ $\displaystyle \delta_{m 0}$ Definition of Unsigned Stirling Numbers of the First Kind

Thus $P \left({0}\right)$ has been shown 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 for all $r \ge 0$, then it logically follows that $P \left({r + 1}\right)$ is true.

So this is the induction hypothesis:

$\displaystyle \forall m \in \Z_{\ge 0}: \sum_k \left\{ {r \atop k}\right\}\left[{k \atop m}\right] \left({-1}\right)^{r - k} = \delta_{m r}$

from which it is to be shown that:

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

### Induction Step

This is the induction step:

 $\displaystyle$  $\displaystyle \sum_k \left\{ { {r + 1} \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k}$ $\displaystyle$ $=$ $\displaystyle \sum_k \left({k \left\{ {r \atop k}\right\} + \left\{ {r \atop k - 1}\right\} }\right) \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k}$ Definition of Stirling Numbers of the Second Kind $\displaystyle$ $=$ $\displaystyle \sum_k k \left\{ {r \atop k}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k} + \sum_k \left\{ {r \atop k - 1}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k}$ $\displaystyle$ $=$ $\displaystyle \sum_k k \left\{ {r \atop k}\right\} \frac 1 k \left({\left[{k + 1 \atop m}\right] - \left[{k \atop m - 1}\right]}\right) \left({-1}\right)^{r + 1 - k}$ Definition of Unsigned Stirling Numbers of the First Kind $\displaystyle$  $\displaystyle + \sum_k \left\{ {r \atop k - 1}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k}$ $\displaystyle$ $=$ $\displaystyle \sum_k \left\{ {r \atop k}\right\} \left[{k + 1 \atop m}\right] \left({-1}\right)^{r + 1 - k} - \sum_k \left\{ {r \atop k}\right\} \left[{k \atop m - 1}\right] \left({-1}\right)^{r + 1 - k}$ $\displaystyle$  $\displaystyle + \sum_k \left\{ {r \atop k - 1}\right\} \left[{k \atop m}\right] \left({-1}\right)^{r + 1 - k}$ $\displaystyle$ $=$ $\displaystyle \sum_k \left\{ {r \atop k}\right\} \left[{k + 1 \atop m}\right] \left({-1}\right)^{r + 1 - k} - \sum_k \left\{ {r \atop k}\right\} \left[{k \atop m - 1}\right] \left({-1}\right)^{r + 1 - k}$ $\displaystyle$  $\displaystyle + \sum_k \left\{ {r \atop k}\right\} \left[{k + 1 \atop m}\right] \left({-1}\right)^{r + 1 - k + 1}$ Translation of Index Variable of Summation $\displaystyle$ $=$ $\displaystyle \sum_k \left\{ {r \atop k}\right\} \left[{k + 1 \atop m}\right] \left({-1}\right)^{r + 1 - k} - \sum_k \left\{ {r \atop k}\right\} \left[{k \atop m - 1}\right] \left({-1}\right)^{r + 1 - k}$ $\displaystyle$  $\displaystyle - \sum_k \left\{ {r \atop k}\right\} \left[{k + 1 \atop m}\right] \left({-1}\right)^{r + 1 - k}$ $\displaystyle$ $=$ $\displaystyle \sum_k \left\{ {r \atop k}\right\} \left[{k \atop m - 1}\right] \left({-1}\right)^{r - k}$ Simplification $\displaystyle$ $=$ $\displaystyle \delta_{\left({m - 1}\right) r}$ Induction Hypothesis $\displaystyle$ $=$ $\displaystyle \delta_{m \left({r + 1}\right)}$ Definition of Kronecker Delta

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

Therefore:

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

$\blacksquare$