This is a very vague question and not particularly related to anything in general, but any thoughts would be helpful.
Say you have permutations $\pi_1,\dots,\pi_n$ of $[2n+1]$. What are some properties that these permutations must have?
For example, you can say that there must be some $x\in[2n+1]$ which never appears first nor last in any of the $\pi_i$'s, i.e. $\pi_i(x)\notin\{1,2n+1\}$ for all $i$. Of course, this can be made more general by saying that for any sets $S_i\subseteq[2n+1]$ with $\sum_i|S_i|\leq 2n$, there is some $x\in[2n+1]$ such that $\pi_i(x)\notin S_i$ for all $i$.
As I said, this is a very vague question. I'm not looking for anything in particular, but if you have any thoughts that you'd like to share, I'd love to hear them! Even if your thought/observation is obvious or silly, I'm still interested in hearing it (afterall, you saw how silly my above observation was).
Thanks :)