Question 43
Let $x_{1}, x_{2}, \ldots, x_{k}$ be a sequence of integers. A rearrangement of this sequence (the numbers in the sequence listed in some other order) is called a scramble if no number in the new sequence is equal to the number originally in its location. For example, if the original sequence is $1,3,3,5$ then $3,5,1,3$ is a scramble, but $3,3,1,5$ is not.
A rearrangement is called a two-two if exactly two of the numbers in the new sequence are each exactly two more than the numbers that originally occupied those locations. For example, $3, 5, 1, 3$ is a two-two of the sequence $1,3,3,5$ (the first two values $3$ and $5$ of the new sequence are exactly two more than their original values $1$ and $3$).
Let $n \geq 2$. Prove that the number of scrambles of
$$
1,1,2,3, \ldots, n-1, n
$$
is equal to the number of two-twos of
$$
1,2,3, \ldots, n, n+1 .
$$
(Notice that both sequences have $n+1$ numbers, but the first one contains two $1$s.)
A rearrangement is called a two-two if exactly two of the numbers in the new sequence are each exactly two more than the numbers that originally occupied those locations. For example, $3, 5, 1, 3$ is a two-two of the sequence $1,3,3,5$ (the first two values $3$ and $5$ of the new sequence are exactly two more than their original values $1$ and $3$).
Let $n \geq 2$. Prove that the number of scrambles of
$$
1,1,2,3, \ldots, n-1, n
$$
is equal to the number of two-twos of
$$
1,2,3, \ldots, n, n+1 .
$$
(Notice that both sequences have $n+1$ numbers, but the first one contains two $1$s.)