Pattern avoidance and quasisymmetric functions

Zachary Hamaker,Brendan Pawlowski,Bruce E Sagan

doi:10.5802/alco.96

Abstract

Given a set of permutations Pi, let S_n(Pi) denote the set of permutations in the symmetric group S_n that avoid every element of Pi in the sense of pattern avoidance. Given a subset S of {1,...,n-1}, let F_S be the fundamental quasisymmetric function indexed by S. Our object of study is the generating function Q_n(Pi) = sum F_{Des sigma} where the sum is over all sigma in S_n(Pi) and Des sigma is the descent set of sigma. We characterize those Pi contained in S_3 such that Q_n(Pi) is symmetric or Schur nonnegative for all n. In the process, we show how each of the resulting Pi can be obtained from a theorem or conjecture involving more general sets of patterns. In particular, we prove results concerning symmetries, shuffles, and Knuth classes, as well as pointing out a relationship with the arc permutations of Elizalde and Roichman. Various conjectures and questions are mentioned throughout.

Highlights

Let Sn be the symmetric group of all permutations of [n] = {1, . . . , n}
We say that permutation σ contains permutation π as a pattern if there is a subsequence σ of elements of σ such that std σ = π
We denote by QSymn the vector space of quasisymmetric functions homogeneous of degree n

Summary

Introduction

Pattern avoidance, quasisymmetric function, Schur function, shuffle, symmetric function, Young tableau. We denote by QSymn the vector space of quasisymmetric functions homogeneous of degree n Bases for this vector space are indexed by compositions of n, which are sequences of positive integers α = To combine permutation patterns and quasisymmetric functions, recall that the descent set of a permutation π = π1 . We provide a citation of their work in the corresponding proposition of this article In this paper, they find Qn(Π) for various sets of patterns not consideered here, for example when Π = {321, 2143, 2413}

Symmetries and the Robinson–Schensted bijection

Shuffles

Partial shuffles

Knuth classes

Arc permutations

Comments and open questions