We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. We directly enumerate the involutions avoiding \(1342\) and the involutions avoiding \(2341\). As we demonstrate, the numerical data for these problems exhibits some surprising behavior. This strange behavior even provides some very unexpected data related to the number of \(1324\)-avoiding permutations.
We are very grateful to Michael Albert for adding support for involutions to his PermLab package.