Forest Polynomials and Pattern Avoidance

TL;DR

The paper identifies exactly when a Schubert polynomial $\mathfrak{S}_w$ coincides with a forest polynomial by pattern avoidance: $\mathfrak{S}_w$ is a forest polynomial if and only if $w$ avoids the six patterns $\{1432, 2413, 2431, 14523, 32154, 341265\}$. It builds a combinatorial bridge between forest polynomials and Schubert polynomials using labeled binary forests, Lehmer codes, and pipe dreams, defining a weight-preserving injection $\psi_w$ from valid forest labelings to pipe dreams of $w$. The proof has a two-direction structure: (i) if $w$ contains a forbidden pattern, surjectivity fails and the polynomials cannot coincide, and (ii) if $w$ avoids the patterns, the injection is surjective and bijective on the corresponding indexing sets. The results connect pattern-avoidance phenomena in permutations to the intersection of two rich polynomial families, and leverage pipe-dream ladder-move dynamics and the Gao theorem for $1432$ to establish the characterization.

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.

Figures (3)

• Figure 1: A binary indexed forest $F$.
• Figure 2: Forest labeling $f_F$ given by $\rho_F$ for $F$ from Figure \ref{['fig:example_forest']}
• Figure 3: Options for forest labelings of $f_F$ of forest $F$

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Example 2.3
• Definition 2.4
• Definition 2.6
• Example 2.7
• Remark 2.8
• Theorem 2.9