Permutations and Pairs of Dyck Paths

Marilena Barnabei,Matteo Silimbani,Flavio Bonetti

doi:10.1155/2013/107454

Abstract

We define a map v between the symmetric group Sn and the set of pairs of Dyck paths of semilength n. We show that the map v is injective when restricted to the set of 1234-avoiding permutations and characterize the image of this map.

Highlights

We say that a permutation σσ σ σσnn contains a pattern ττ τττkk if σσ contains a subsequence that is order-isomorphic to ττ
E sets of permutations that avoid a single pattern ττ τ SS3 have been completely determined in last decades
We present a bijection between SSnn(1234) and a set of pairs of Dyck paths of semilength nn

Summary

Introduction

We say that a permutation σσ σ σσnn contains a pattern ττ τττkk if σσ contains a subsequence that is order-isomorphic to ττ. Denote by SSnn(τττ the set of permutations in SSnn avoiding ττ. E sets of permutations that avoid a single pattern ττ τ SS3 have been completely determined in last decades. It has been shown [1] that, for every ττ τττ, the cardinality of the set SSnn(τττ equals the nnth Catalan number, which is the number of Dyck paths of semilength nn (see [2] for an exhaustive survey). Many bijections between SSnn(τττ, ττ τττ, and the set of Dyck paths of semilength nn have been described (see [3] for a fully detailed overview). We de ne a map νν from SSnn to the set of pairs of Dyck paths, prove that every element in the image of νν corresponds to a single element in SSnn(1234), and characterize the set of all pairs that belong to the image of the map νν

Dyck Paths

LTR Minima and RTL Maxima of a Permutation

The Maps λλ and μμ

Main Results