# Zero-one Schubert polynomials

## TL;DR

This paper explores the structure of Schubert polynomials, showing how they relate to pattern containment and characterizing those with coefficients only 0 or 1, linking them to generalized permutahedra.

## Contribution

It introduces a pattern avoidance characterization for zero-one Schubert polynomials and connects these polynomials to integer point transforms of generalized permutahedra.

## Key findings

- Schubert polynomials with 0-1 coefficients are pattern-closed.
- Characterization of such polynomials via twelve avoided patterns.
- Zero-one Schubert polynomials correspond to integer point transforms of generalized permutahedra.

## Abstract

We prove that if $\sigma \in S_m$ is a pattern of $w \in S_n$, then we can express the Schubert polynomial $\mathfrak{S}_w$ as a monomial times $\mathfrak{S}_\sigma$ (in reindexed variables) plus a polynomial with nonnegative coefficients. This implies that the set of permutations whose Schubert polynomials have all their coefficients equal to either 0 or 1 is closed under pattern containment. Using Magyar's orthodontia, we characterize this class by a list of twelve avoided patterns. We also give other equivalent conditions on $\mathfrak{S}_w$ being zero-one. In this case, the Schubert polynomial $\mathfrak{S}_w$ is equal to the integer point transform of a generalized permutahedron.