I've gotten stuck on a slightly random combinatorial question, and I'm doing a bit of a shot in the dark here to see if someone else has thoughts about it. I'm interested in studying a permutation of an odd number of elements $\{1,\dots, 2n+1\}$ which preserves parity, i.e. it sends odd numbers to odd numbers and even numbers to even numbers. What I want to show is that such a permutation has at least as many inversions of pairs with opposite parity as it does of pairs with the same parity.
"Please solve this conjecture" isn't a good MathOverflow question, so let me try to summon a better one:
Does anyone know of any tools for studying the structure of inversion sets of permutations that would be useful for keeping track of parity like this?
I feel like there's some natural approach to this I'm missing. One obvious possibility would be if there was some natural factorization compatible with the length function i.e. a factorization $\sigma=\sigma'\sigma''$ where $\ell(\sigma)=\ell(\sigma')+\ell(\sigma'')$. The problem is that it’s hard to do this while keeping $\sigma'$ and $\sigma''$ parity preserving. For example $(13)(24)$ has no factorization like this.
