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## permutation

**Warning: This is a draft! Has a lot of errors and typos**

Consider an element in $S_4$

$$
\sigma(1)=4,\qquad \sigma(2)=2\qquad \sigma(3)=1,\qquad \sigma(4)=3
$$

**How many order-reversing paris are there?** What are they

$$(1,2)\mapsto(4,2)$$

$$(1,3)\mapsto(4,1)$$

$$(1,4)\mapsto(4,3)$$

$$(2,3)\mapsto(2,1)$$

$$
\sigma(2)=2,\sigma(3)=1
$$

$$
\sigma(3)=1,\sigma(2)=2
$$


$$
1\mapsto 4\mapsto 3\mapsto 1
$$

$$
2\mapsto 2
$$

#### Let us write this $\sigma$ into a permuted list, then let's see what is the order-reversing pairs mean


$$[4,2,1,3]$$

decreasing sub-pairs: $(4,2),(4,1),(4,3),(2,1)$

That means, the permutation corresponding to $4,2,1,3$ is an even permutation.


### Now let us change two elements in the list and let's see how the parity update

#### I want to change the position of 2 and 3


**Old permuted list**: 4,2,1,3

**New permuted list**: 4,3,1,2

**To see how the parity change**, we wish to know how the decreasing sub-pairs change.

**Classify original sub-pairs into the following classes**



$(4,2)$, $(4,3)$: sub-pair that 4 is not an affected number, 2 is affected number, and 4 is **not between** affected numbers 2 and 3. <span style="color:red"> the order would not change</span>

$(4,1)$: sub-pair that does not have any affected numbers. <span style="color:red">do not change the order</span>


$(2,1)$: sub-pair that has one affected number, but the other number is **between two affected numbers**. Then **It's order would change definitely** $(2,1)\mapsto(1,2)$ <span style="color:red">it changes the order!!!</span>

$(1,3)$: sub-pair that has one affected number, but the other number is **between two affected numbers**. Then **It's order would change definitely** $(1,3)\mapsto(3,1)$ <span style="color:red">it changes the order!!!</span>



$(2,3)$ : sub-pair that both numbers are affected it must change its order
$(2,3)\mapsto (3,2)$










### Excercise

Now think about if I have a permuted list
$$
(1,3,2,5,7,8,9,6,4)
$$

I want to change the $3,6$, after I change it, it become
$$
(1,6,2,5,7,8,9,3,4)
$$

Tell me : How many sub-pairs has changed its order.



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## Conclusion 

After I switch any two elements of the permuted list, the parity of the permutation MUST CHANGE

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### Let's see

Previously we know that $(4,2,1,3)$ is an even permutation by counting how counting how many decreasing paris are there, now let us use the easiest method to do so.

$$\sigma(1)=4$$

$$\sigma(2)=2$$

$$\sigma(3)=1$$

$$\sigma(4)=3$$

Let's just draw the cycles

$$
1\mapsto 4\mapsto 3\mapsto 1\text{ cycle of size 3}
$$

$$
2\mapsto 2 \text{ cycle of size 1}
$$

How many even-sized cycle are there? 0-- Even number . So it is an even permutation.

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### Excercise

For the permuted list as the following

$$(3,4,1,7,8,2,5,9,6)$$

Is this corresponds to an odd or even permutation?

What are sizes of the cycles?

Two cycles, 1 cycle of size 2, another cycle of size 7

$$\sigma(1)=3$$

$$\sigma(2)=4$$

$$\sigma(3)=1$$

$$\sigma(4)=7$$

$$\sigma(5)=8$$

$$\sigma(6)=2$$

$$\sigma(7)=5$$

$$\sigma(8)=9$$

$$\sigma(9)=6$$

$$
1\mapsto 3 \mapsto 1 \text{ cycle of size 2}
$$

$$
4\mapsto 7\mapsto 5\mapsto 8\mapsto 9\mapsto 6\mapsto 2\mapsto 4 \text{ cycle of size 7}
$$


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Let 's see why switching two rows will switch the sign of $f$

$$
f\begin{pmatrix}
{\color{red}1}&2&3\\4&5&{\color{red}6}\\7&{\color{red}8}&9\\
\end{pmatrix}
$$

Tell me which permutation is corresponding to this selection of elements. $1\times 6\times 8$

This is corresponding to permutation $(1,3,2)$

$$
f\begin{pmatrix}
4&5&{\color{red}6}\\{\color{red}1}&2&3\\7&{\color{red}8}&9\\
\end{pmatrix}
$$

In this, tell me which permutation is corresponding to the term $6\times 1\times 8$

$(3,1,2)$


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