Given a sequence $\vec{\sigma}$ of length $L$ and two given elements $i$ and $j$ with $i$ before $j$. We want to look at all possible even permutation of $\vec{\sigma}$ and see how many of them has $i$ stays before $j$.
I have a feeling that we can solve this by looking at the even permutation group and the right coset of the subgroup {1, (i j)}, but cannot figure out the details.