# Construction of Permutations/Proof 2

## Contents

## Theorem

The ${}^n P_n$ permutations of $n$ objects can be generated algorithmically.

By Number of Permutations, that number is given by:

- ${}^n P_n = n!$

where $n!$ denotes the factorial of $n$.

This will be demonstrated to hold.

## Proof

The following is an inductive method of creating all the permutations of $n$ objects.

### Base Case

There is clearly one way to arrange one object in order.

### Inductive Hypothesis

We assume that we have constructed all $n!$ permutations of $n$ objects.

### Induction Step

Without loss of generality, let a set $S_n$ of $n$ objects be $\left\{{1, 2, \ldots, n}\right\}$.

Take a permutation of $S_n$:

- $a_1 \, a_2 \, a_3 \, \ldots \, a_n$

Construct the array:

- $a_1 \, a_2 \, a_3 \, \ldots \, a_n \, \dfrac 1 2, \quad a_1 \, a_2 \, a_3 \, \ldots \, a_n \, \dfrac 3 2, \quad a_1 \, a_2 \, a_3 \, \ldots \, a_n \, \dfrac 5 2, \quad \ldots, \quad a_1 \, a_2 \, a_3 \, \ldots \, a_n \, \left({n + \dfrac 1 2}\right)$

This will contain $n + 1$ objects.

Now rename the elements of each permutation using the numbers $\left\{ {1, 2, \ldots, n}\right\}$ preserving order.

It is clear that all permutations of $n + 1$ objects can be obtained in this manner, and no permutation is obtained more than once.

As there are ${}^n P_n$ permutations on $n$ objects, there are $\left({n + 1}\right) \times {}^n P_n$ permutations on $n + 1$ objects.

Hence by induction, and the recursive definition of the factorial:

- ${}^n P_n = n!$

$\blacksquare$

## Example

From the permutation of $4$ objects $3 1 2 4$, the following permutations of $5$ objects can be made as follows:

- $3 1 2 4 \dfrac 1 2$
- $3 1 2 4 \dfrac 3 2$
- $3 1 2 4 \dfrac 5 2$
- $3 1 2 4 \dfrac 7 2$
- $3 1 2 4 \dfrac 9 2$

Renaming according to size:

- $4 2 3 5 1$
- $4 1 3 5 2$
- $4 1 2 5 3$
- $3 1 2 5 4$
- $3 1 2 4 5$

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials