Let $\mathfrak{S}_{2n}$ be the permutation group of the letters $[2n]=\{1,2,\dots,2n\}$. Call a permutation $\pi\in\mathfrak{S}_{2n}$ has an *$n$-distant pair* if there is some $j\in [2n-1]$ such that $\vert\pi_j-\pi_{j+1}\vert=n$.

Define $A=$ set of $\pi\in\mathfrak{S}_{2n}$ with exactly one $n$-distant pair and $B=$ set of permutations without $n$-distant pairs.

Question.Which inequality is true in general: $\vert A\vert>\vert B\vert$ or $\vert A\vert<\vert B\vert$? And your proof?