Question

What is a permutation?

Answer 1

A bijection from the natural numbers to itself. Basically, its a reordering.

Answer 2

I've seen some definitions, but I want to write one by myself.

Answer 3

Ahh that's is interesting, where did you get that definition |malevolence19?

Answer 4

I've reading college's algebra books and I found this: I'm think is the most complete: "A permutation is an ordered arrangement of r objects chosen from n objects".

Answer 5

I think the definition that uses the bijection concept does not take account of the r objects, from n

Answer 6

So, we can think of it as a function which takes a set of positive integers and maps it to itself (in a different order or not). A permutation would be like:
\[\sigma=<1,3>\]
This notation is called cycle notation. It means it sends 1 to 3 and 3 to 1. So if we had:
\[<1,2,3,4,...,n-1,n>\]
And applied sigma to it, we would get:
\[<3,2,1,4,...,n-1,n>\]
Since everything else is sent to itself. Permutations are useful for things such as defining the determinant of a matrix:
Let A be a nxn matrix then:
\[\det(A)=\sum_{\sigma \in \S_n}\left[ sgn(\sigma) \prod_{k=1}^{n}a_{k \sigma(k)} \right]\]
But thats a whole different story.

That is the definition I learned.

Answer 7

But a permutation itself is a function. So a permutation isn't a set of objects, its a rearrangement.

Answer 8

Hey, I was reading your answer and suddendly dissapeared, what happened? @across

Answer 9

yeah, I remember that definition of determinant.

Answer 10

Ok, So How can I write this permutation with that notation? [1,2,3,4} --> {3,2,4,1}? \[\sigma<1,3>\sigma<3,4>\sigma<4,1>\]

Answer 11

Am I wrong?

Answer 12

You can do this:
\[\sigma=<1,3,4>\]
Sends 1 to 3, 3 to 4, then 4 to 1. In general:
\[\sigma <i_1,i_2,...,i_n,...,i_{k-1},i_k>\]
Sends:
\[i_n \rightarrow i_{n+1}; i_k \rightarrow i_1\]

Answer 13

Ok, thank you malevolence!