Counting occurrences of 231 in an involution

Abstract

We study the generating function for the number of involutions on n letters containing exactly r ≥ 0 occurrences of 3412. It is shown that finding this function for a given r amounts to a routine check of all involutions on 2r + 1 letters.

Decomposing simple permutations, with enumerative consequences

“…• Involutions in Av(231 ≤r ) -Mansour, Yan, and Yang [18]. In the same reference: even involutions in Av(231 ≤r ).…”

Restricted Symmetric Permutations

“…It is an easy exercise to show that D 8 has exactly ten subgroups. Here we list these subgroups, using subscripts with the same numerical part for conjugate subgroups and writing e to denote the identity in D 8 . 8 .…”

“…Here we list these subgroups, using subscripts with the same numerical part for conjugate subgroups and writing e to denote the identity in D 8 . 8 . The permutations which are invariant under H 4a (that is, the involutions) and avoid various sets of patterns have been widely studied (a small sample of the literature includes [2,5,6,8,11]), but pattern-avoiding permutations which are invariant under the other non-identity subgroups of D 8 have received little or no enumerative attention.…”

Symmetric Permutations Avoiding Two Patterns

“…D 4 , the group of the eight symmetries of a square, is well-known to be generated by the above mappings r, c, and i. We say that a permutation is preserved under some symmetry g ∈ D 4 if its diagram is unchanged by g. Equivalently, if we consider D 4 to be a group of actions on the set of diagrams of permutations in S n , then π ∈ S n is preserved by g ∈ D 4 if g is in the stabilizer of the diagram of π. Since the stabilizer of a diagram is a subgroup of D 4 , we can consider the possible symmetries of a permutation by considering the 10 distinct subgroups of D 4 .…”

“…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 .…”