Symmetric Permutations Avoiding Two Patterns

Abstract

Symmetric pattern-avoiding permutations are restricted permutations which are invariant under actions of certain subgroups of D 4 , the symmetry group of a square. We examine pattern-avoiding permutations with 180 • rotational-symmetry. In particular, we use combinatorial techniques to enumerate symmetric permutations which avoid one pattern of length three and one pattern of length four. Our results involve well-known sequences such as the alternating Fibonacci numbers, Catalan numbers, triangular numbers, and powers of two.

On the centrosymmetric permutations in a class

“…We are concerned with the number of centrosymmetric permutations in a class C. Past research has focused on finding this number for specific classes C. Egge [11] found the number of centrosymmetric permutations in Av n (R) for every set R of size-3 permutations. Lonoff and Ostroff [17] did the same when R consists of one size-3 and one size-4 permutation. Egge [12] found an expression for Av n (k .…”

Pattern Avoidance in Reverse Double Lists

“…Pattern-avoidance in centrosymmetric permutations, i.e. permutations π such that π rc " π has been studied by Egge (2007Egge ( , 2010, by Lonoff and Ostroff (2010), and by Barnabei et al (2010). Ferrari (2011) generalized this idea to pattern avoidance in centrosymmetric words.…”

Enumerating rc-Invariant Permutations with No Long Decreasing Subsequences

“…Often these results include refined enumerations, whose answers are given naturally in terms of binomial coefficients. Building on Egge's work, Lonoff and Ostroff [10] have enumerated S H n (R) for H = {e, rc} and almost all R consisting of one pattern of length 3 and one pattern of length 4. Their answers include the Fibonacci numbers, powers of 2, perfect squares, and triangular numbers, and their work has led to a Fibonacci identity which appears to be new.…”