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 D 8 , the symmetry group of a square. Three other nontrivial subgroups of D 8 also have invariant 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 S 3 . For each subgroup we also give a bijection between the invariant permutations and certain symmetric signed permutations.

Enumerating rc-Invariant Permutations with No Long Decreasing Subsequences

“…The group D 4 acts naturally on these diagrams, and therefore on permutations; for any g ∈ D 4 we write π g to denote the image of π under g. We observe that π ∈ S n contains σ ∈ S k if and only if there exist k rows and k columns in the diagram of π whose squares of intersection form the diagram of σ. It follows that if g ∈ D 4 then π avoids σ if and only if π g avoids σ g .…”

“…It follows that if g ∈ D 4 then π avoids σ if and only if π g avoids σ g . In this context it is traditional to single out the following three elements of D 4 .…”

“…For any subgroup H ⊆ D 4 and any set R of permutations, let S H n (R) denote the set of permutations in S n which are invariant under every element of H and avoid every pattern in R. The set S H n (R) has been studied for a variety of R when H = {e} or H = {e, i}; the first results along these lines are those of Simion and Schmidt [15]. More recently, Egge [4] has enumerated S H n (R) for other subgroups H ⊆ D 4 when R consists of relatively small patterns. Often these results include refined enumerations, whose answers are given naturally in terms of binomial coefficients.…”

Symmetric Permutations Avoiding Two Patterns

“…For instance, for n ≥ 2, no permutation in S n is preserved by r, since the first and last elements in one-line notation are never equal. There are four subgroups which are interesting to study, and have been to various extents (a sample of the literature includes [2,3,4,5]). In this paper we will focus our attention on the subgroup {e, rc}, and label the set of permutations of length n preserved by this subgroup S rc n .…”

“…For instance, 412563 ∈ S rc 6 . Just as in S n , we use S rc n (R) to refer to the set of permutations of length n whose diagrams are symmetric under 180 • rotation and which avoid every pattern in R. These sets have been enumerated for certain R by Egge [3], in particular for all R ⊆ S 3 . In this paper we will enumerate S rc n (σ, τ ) for all σ ∈ S 3 and τ ∈ S 4 .…”