Zero-one Schubert polynomials

2019/03/25 by Fink, Alex, Mészáros, Karola, Dizier, Avery St.
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.10332

Abstract

We prove that if σ∈ S_m is a pattern of w ∈ S_n, then we can express the Schubert polynomial \mathfrakS_w as a monomial times \mathfrakS_σ (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 \mathfrakS_w being zero-one. In this case, the Schubert polynomial \mathfrakS_w is equal to the integer point transform of a generalized permutahedron.

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• Zero-one Schubert polynomials
• Schubert Polynomials and Elementary Symmetric Products
• Zero-one Grothendieck Polynomials
• Single-SEM Schubert Polynomials
• Schubert polynomials and patterns in permutations