In October 2010, I published a Monthly problem that introduced the concept of a fair permutation, which is a permutation $\pi$ such that for every $i$, either $\pi(i) > i$ and $\pi^{-1}(i) > i$, or $\pi(i) \le i$ and $\pi^{-1}(i) \le i$. Equivalently, every cycle of $\pi$ is either a fixed point or an alternating cycle of even length, meaning that if $z$ is the largest element of the cycle, then $$z > \pi(z) < \pi(\pi(z)) > \pi(\pi(\pi(z))) < \cdots < z.$$

Now I can show by generatingfunctionology that the number of fair permutations is twice the number of Salié permutations. A permutation $\sigma$ on $\lbrace1, 2, \ldots, 2m\rbrace$ is said to be a Salié permutation if for some $r\le m$, $$\sigma(1) < \sigma(2) > \sigma(3) < \cdots < \sigma(2r)$$ and $$\sigma(2r) < \sigma(2r+1) < \sigma(2r+2) < \cdots < \sigma(2m).$$

Question: Can one construct an explicit 2-to-1 map from fair permutations to Salié permutations?