Forest Polynomials and Pattern Avoidance

TL;DR

This paper explores forest polynomials as a quasisymmetric analogue of Schubert polynomials, revealing that Schubert polynomials decompose into forest polynomials precisely when certain pattern avoidance conditions are met.

Contribution

It characterizes Schubert polynomials that are forest polynomials through pattern avoidance of six specific permutations, linking pattern avoidance to polynomial decomposition.

Findings

Schubert polynomials decompose positively into forest polynomials

A Schubert polynomial is a forest polynomial if and only if its permutation avoids six patterns

Pattern avoidance controls key properties of Schubert polynomials

Abstract

Forest polynomials, recently introduced by Nadeau and Tewari, can be thought of as a quasisymmetric analogue for Schubert polynomials. They have already been shown to exhibit interesting interactions with Schubert polynomials; for example, Schubert polynomials decompose positively into forest polynomials. We further describe this relationship by showing that a Schubert polynomial $\mathfrak{S}_w$ is a forest polynomial exactly when $w$ avoids a set of $6$ patterns. This result adds to the long list of properties of Schubert polynomials that are controlled by pattern avoidance.