I've been working on sorting and factorisation problems on permutations for some time now, and have observed that given a permutation $\pi$ of $n$ elements, the permutation $\pi^\chi=\chi\circ\pi\circ\chi^{-1}$, where $\chi=\chi^{-1}=(n\ n-1\ \cdots\ 2\ 1)$, often has attractive properties (with respect to a particular sorting problem).


Is there a name for this "special" permutation (other than "the conjugate of $\pi$ by $\chi$")?



Name of a particular conjugate permutation
This is the reverse-complement of $\pi$. 


In one-line notation, the reverse of a permutation is what you get by writing it backwards and the complement of a permutation is what you get when you replace each entry $i$ by $n -i + 1$. (In other words, one of these operations is multiplication by $\chi$ on the right, the other on the left.) The reverse-complement is what you get by doing both of these operations, or equivalently by giving the permutation matrix a half-turn. (Together with inversion, these operations generate the dihedral group acting on each permutation matrix.)


