Permutations (14.1-14.3)

Prove that, by successively transposing adjacent numbers, any two elements of a permutation can be interchanged while preserving the arrangement of the remaining elements.

Prove that the parity of a permutation changes if two elements in it are interchanged.

Prove that, by transposing adjacent numbers a finite number $(k)$ of times, the elements of a permutation can be arranged in increasing order. Is $k$ uniquely determined?

Let $s$ be the number of inversions in a permutation. Prove that the numbers $k$ and $s$ have the same parity.

Indicate a sequence of $s$ interchanges of pairs of adjacent numbers, as a result of which all the elements of the permutation are arranged in increasing order.

$\left(\begin{array}{lllll}5 & 4 & 3 & 2 & 1\end{array}\right)$

$\left(\begin{array}{llllll}6 & 4 & 5 & 2 & 3 & 1\end{array}\right)$

$\left(\begin{array}{llllll}1 & 2 & 4 & 5 & 6 & 3\end{array}\right)$

$\left(\begin{array}{lllllllll}1 & 2 & 4 & 3 & 5 & 9 & 8 & 7 & 6\end{array}\right)$

$\left(\begin{array}{lllllllll}9 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1\end{array}\right)$

$\left(\begin{array}{lllllllll}4 & 3 & 2 & 1 & 5 & 9 & 8 & 7 & 6\end{array}\right)$

$(n, n - 1, \ldots , 1)$

$(1, 3, 5, \ldots , 2n - 1, 2, 4, 6, \ldots , 2n)$

$(2, 4, 6, \ldots , 2n, 1, 3, 5, \ldots , 2n + 1)$.