Question 34
Let $n$ be a positive integer divisible by 4. Consider permutations $(a_1,a_2,\ldots,a_n)$ of $(1,2,\ldots,n)$ with the following property: for every $j$, if $i=a_j$, then $a_i+j=n+1$. Prove that there are exactly $\dfrac{(n/2)!}{(n/4)!}$ such permutations.