This question is motivated by the definition of pfaffian.
Let $n$ be a nonnegative integer. Let $\Pi$ be the subset of the symmetric group $S_{2n}$ consisting of those permutations $\pi$ that can be written in the form $$ \pi=\left[\begin{array}{ccccccc} {1} & {2} & {3} & {4} & {\cdots} & {2 n-1} & {2 n} \\ {i_{1}} & {j_{1}} & {i_{2}} & {j_{2}} & {\cdots} & {i_{n}} & {j_{n}} \end{array}\right] $$ (two-line notation) with $i_{k}<j_{k}, \forall k$, and $i_{1}<i_{2}<\cdots<i_{n}$.
How to compute the number $\sum_{\pi \in \Pi} \operatorname{sgn}\left(\pi\right) $?
Note that the elements of $\Pi$ are precisely the permutations $\pi_\alpha$ in https://en.wikipedia.org/wiki/Pfaffian#Formal_definition .