Permutations with Unique Fixed and Reflected Points

We examine permutations having a unique fixed point and a unique reflected point; such permutations correspond to permutation matrices having exactly one \(1\) on each of the two main diagonals. The permutations are of two types according to whether or not the fixed point is the same as the reflected point. We also consider permutations having no fixed or reflected points; these have been enumerated using two different methods, and we employ one of these to count permutations with unique fixed and reflected points.