Restricted Symmetric Permutations

Abstract

Pattern-avoiding involutions, which have received much enumerative attention, are pattern-avoiding permutations which are invariant under the natural action of a certain subgroup of D8, the symmetry group of a square. Three other nontrivial subgroups of D8 also have in- variant permutations under this action. For each of these subgroups, we enumerate the set of permutations which are invariant under the action of the subgroup and which also avoid a given set of forbidden patterns. The sets of forbidden patterns we consider include all subsets of S3. For each subgroup we also give a bijection between the invariant permutations and certain symmetric signed permutations.