A permutation
![\{x_1, \ldots, x_{2n}\}](/media/m/f/e/9/fe933476f3873023f94fa753f8cd9ed6.png)
of the set
![\{1,2, \ldots, 2n\}](/media/m/f/8/b/f8b469a2870e43918be681f2561e1655.png)
where
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
is a positive integer, is said to have property
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
if
![|x_i - x_{i + 1}| = n](/media/m/3/f/4/3f4dc57a315068855e6e7a3677b2e0e0.png)
for at least one
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
in
![\{1,2, \ldots, 2n - 1\}.](/media/m/0/5/b/05b64af4faec32f001e96249272a2f65.png)
Show that, for each
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
, there are more permutations with property
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
than without.
%V0
A permutation $\{x_1, \ldots, x_{2n}\}$ of the set $\{1,2, \ldots, 2n\}$ where $n$ is a positive integer, is said to have property $T$ if $|x_i - x_{i + 1}| = n$ for at least one $i$ in $\{1,2, \ldots, 2n - 1\}.$ Show that, for each $n$, there are more permutations with property $T$ than without.