# Sum over k to n of Unsigned Stirling Number of the First Kind of k with m by n factorial over k factorial

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## Contents

## Theorem

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

- $\displaystyle \sum_{k \mathop \le n} {k \brack m} \frac {n!} {k!} = {n + 1 \brack m + 1}$

where:

- $\displaystyle {k \brack m}$ denotes an unsigned Stirling number of the first kind
- $ n!$ denotes a factorial.

## Proof

The proof proceeds by induction on $n$.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:

- $\displaystyle \forall n \in \Z_{\ge 0}: \sum_{k \mathop \le n} {k \brack m} \frac {n!} {k!} = {n + 1 \brack m + 1}$

### Basis for the Induction

$\map P 0$ is the case:

\(\displaystyle \) | \(\) | \(\displaystyle \sum_{k \mathop \le 0} {k \brack m} \frac {0!} {k!}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle {0 \brack m} \frac 1 {0!}\) | Definition of Unsigned Stirling Numbers of the First Kind | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \delta_{0 m}\) | Unsigned Stirling Number of the First Kind of 0 | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \delta_{1 \paren {m + 1} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle {1 \brack m + 1}\) | Unsigned Stirling Number of the First Kind of 1 |

So $\map P 0$ is seen to hold.

This is the basis for the induction.

### Induction Hypothesis

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

So this is the induction hypothesis:

- $\displaystyle \sum_{k \mathop \le r} {k \brack m} \frac {r!} {k!} = {r + 1 \brack m + 1}$

from which it is to be shown that:

- $\displaystyle \sum_{k \mathop \le r + 1} {k \brack m} \frac {\paren {r + 1}!} {k!} = {r + 2 \brack m + 1}$

### Induction Step

This is the induction step:

\(\displaystyle \) | \(\) | \(\displaystyle \sum_{k \mathop \le r + 1} {k \brack m} \frac {\paren {r + 1}!} {k!}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle {r + 1 \brack m} \frac {\paren {r + 1}!} {\paren {r + 1}!} + \paren {r + 1} \paren {\sum_{k \mathop \le r} {k \brack m} \frac {r!} {k!} }\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle {r + 1 \brack m} + \paren {r + 1} {r + 1 \brack m + 1}\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle {r + 2 \brack m + 1}\) | Definition of Unsigned Stirling Numbers of the First Kind |

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

Therefore:

- $\displaystyle \forall m, n \in \Z_{\ge 0}: \sum_{k \mathop \le n} {k \brack m} \frac {n!} {k!} = {n + 1 \brack m + 1}$

$\blacksquare$

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $(56)$