What will be the possible permutations of set S={1,2,3,4}. And how many are of those odd and even? I have solved it as per my knowledge and understanding.
Since there are 4 elements so no. of permutations will be
$$4! = 24$$
$$(1,2,3,4), (1,2,4,3), (1,3,2,4), (1,3,4,2), (1,4,2,3), (1,4,3,2), (2,1,3,4), (2,1,4,3), (2,3,1,4), (2,3,4,1), (2,4,1,3), (2,4,3,1), (3,2,1,4), (3,2,4,1), (3,1,2,4), (3,1,4,2), (3,4,2,1), (3,4,1,2), (4,2,3,1), (4,2,1,3), (4,3,2,1), (4,3,1,2), (4,1,2,3), (4,1,3,2)$$
Is it correct?
• And for even and odd-
Even: product of even no of transpositions. e.g., $(1,2,3) = (1,2)(1,3)$ is even
Odd: product of odd no of transpositions. e.g., $(1,2,3,4) = (1,2)(1,3)(1,4)$ is odd.
• But if I apply this theory in this question then all permutations will be odd.
I am confused here, is my solution incorrect? May be there will be more permutations with less elements like $(1,2,3) , (1,2,4)$?
• Also there is a swapping logic! If the numbers are swapped odd times then it is odd and even otherwise
So in this case $(1,2,3,4)$ is even (no swaps)
$(3,2,1,4)$ = Swap 1<->3 is odd
Also $(4,2,1,3)$ = Swap 3<->4 then swap 1<->4 = even permutation
What will be the answer to my question?

There are twenty four elements in $\mathcal{S}_4$, namely $$\{\operatorname{id}, (1\ 2), (1\ 3), (1\ 4), (2\ 3),\ (2\ 4), (3\ 4), (1\ 2\ 3),\ (1\ 2\ 4), (1\ 3\ 4), (2\ 3\ 4),\ (1\ 3\ 2),\ (1\ 4\ 2),\ (1\ 4\ 3), (2\ 4\ 3),\ (1\ 2\ 3\ 4), (1\ 3\ 4\ 2), (1\ 4\ 2\ 3), (4\ 3\ 2\ 1), (2\ 4\ 3\ 1), (3\ 2\ 4\ 1), (1\ 2)(3\ 4), (1\ 3)(2  \ 4),\ (1\ 4)(2\ 3)\}.$$ In general; there are $n!$ elements in $\mathcal{S}_n.$
Now, list the cycle types of $\mathcal{S}_4$ $$\{(1, 1, 1, 1), (2,1,1),(2,2),(3,1),(4)\}.$$ List how many cycles each cycle type takes, let's call this quantity $\gamma(a)$ where $a$ corresponds to each cycle type. Then $$\gamma(1,1,1,1)=4,\quad\gamma(2,1,1)=3,\quad\gamma(2,2)=2,\quad\gamma(3,1)=2,\quad\gamma(4)=1.$$ Then, a permutation is even if $n-\gamma(a)$ is even. So, the even permutations are the ones of cycle type $(1,1,1,1)$, $(2,2)$, $(3,1)$. This gives us a total of twelve even permutations, namely $$\{\operatorname{id}, (1\ 2)(3\ 4), (1\ 3)(2  \ 4),\ (1\ 4)(2\ 3),(1\ 2\ 3),\ (1\ 2\ 4), (1\ 3\ 4), (2\ 3\ 4),\ (1\ 3\ 2),\ (1\ 4\ 2),\ (1\ 4\ 3), (2\ 4\ 3)\},$$ with the other twelve permutations being odd.
In general, there are $n!/2$ even and odd permutations in $\mathcal{S}_n$.
