What is the connection of these definitions of a Permutation?

Question

sorry if the title is confusing, I didn't know how else to phrase this question.

I know that a permutation is "the number of ways something can be uniquely ordered." The classic example is you have a standard deck of cards, and so there are 52! unique ways to arrange said deck of cards.

But, I have learned that a permutation on a set A is just a bijection of the set A that maps to itself. So, assuming f is bijective,

f: A -> A

I am just confused trying to relate these two concepts and seeking some insight to help me understand this. Thank you!

Answer 1

Let's say we have a deck of 3 cards only; say the ace, two and three of diamonds.

There are also three places in the deck a card may occur: first, second and third.

So, an "arrangement" of the deck corresponds to a mapping from the set of cards, to the set of places the card may occur in. Because these both have the same cardinality (see what I did there?), it's convenient to use the same set, namely: $S = \{1,2,3\}$ to index both sets.

This gives a bijection between two sets of bijections:

Set $1$: $\{\text{set of cards}\} \to \{\text{set of positions}\}$

Set $2: S \to S$.

For example, the arrangment: "ace of diamonds, three of diamonds, two of diamonds" can be thought of as the mapping:

$\text{ace}\mapsto \text{first}\\ \text{two} \mapsto \text{third}\\ \text{three} \mapsto \text{second}$

which then corresponds to the mapping:

$1 \mapsto 1\\ 2\mapsto 3\\ 3 \mapsto 2$

which is often thought of as a transposition, or "swap" of the last two cards. Other possible arrangements are mapped into bijections $S \to S$ in a similar fashion.

The idea is pretty much the same for sets of larger cardinalities (but a bit more tedious to work out).

Answer 2

Think of the “slots” into which elements of $A$ can be placed as being labeled by elements of $A$. For any $a\in A$, then, the function $f$ tells you what’s in the slot labeled “$a$.”