There is a known result that states the following about the number of inversions in a permutation. In particular, a permutation has an odd number of inversions if and only if it contains an odd number of even length cycles.
I would like to know if the number of inversions can be further characterized in the case where there is exactly one even length cycle. More precisely, what can we say about the number of inversions if a permutation of size $2^k$ (where $k$ is a positive integer) consists of exactly one cycle of length $2^k$?
Edit: Perhaps an exact formula may not be known, but there may be other interesting properties that help characterize the number of inversions. I am not very familiar with the literature in this area, so any potential references would be appreciated.