# A permutation with reflection property

Consider permutations $\pi$ of the set $\{1,\dots,n\}$ having the symmetry property $\pi \pi^* \pi = \pi^*$, where $\pi^*$ is the "reflection" $k \mapsto n+1-k$. Are there references or other information about such a $\pi$?

• You really do mean $\pi\pi^*\pi$, not $\pi\pi^*\pi^{-1}$? – LSpice Sep 12 at 19:09
• Also, is $\pi^*$ meant to depend on $\pi$, say as $k \mapsto n + 1 - \pi(k)$, or is it really a fixed permutation independent of $\pi$? If the latter, then you are just looking for permutations conjugate by a fixed involution to their inverses. – LSpice Sep 12 at 19:10
• In other words, $\pi \pi^{*}$ is involution. It should reduce the questions about your permutations to the questions about involutions. – Fedor Petrov Sep 12 at 19:11
• @FedorPetrov's re-phrasing is better than mine. – LSpice Sep 12 at 19:11
• Why don't you use a notation without the letter $\pi$ to denote the reflection? – YCor Sep 15 at 16:59

The problem boils down to finding permutations $\pi$ such that $\pi(n+1-x)=y\iff\pi(n+1-y)=x$. Let $X\subset\{1,...,n\}$ be such that $n-\lvert X\rvert$ is even and let $P$ be a partition of $\{1,...,n\}\backslash X$ into pairs and denote, for every element $k$ of $\{1,...,n\}\backslash X$, its companion by $P(k)$. Then, the permutation defined by $\pi(n+1-k)=P(k)$ for every $k\in\{1,...,n\}\backslash X$ and $\pi(n+1-x)=x$ for every $x\in X$ satisfies the required condition. It is clear that appropriate permutations are in a one-to-one correspondence with $(X,P)$ couples.
• Isn't this just a complicated way of saying "$\pi$ is of the form $\sigma\pi^*$, where $\sigma$ is an involution" (which is what @FedorPetrov's comment's that you quote says)? I guess it depends on the poster's needs which form of the statement is more convenient …. – LSpice Sep 15 at 12:49
• By the way, as a TeX matter, \mid is only intended as a binary operator (like 2 \mid 6: $2 \mid 6$). For delimiters, you want \lvert and \rvert (like \lvert2\rvert: $\lvert2\rvert$). I made the edit here. – LSpice Sep 15 at 17:41