Inclusion-exclusion on Schubert polynomials

TL;DR

This paper proves a nonnegativity property of certain Schubert polynomials for pattern-avoiding permutations, advancing understanding of their structure and partially confirming a recent conjecture.

Contribution

It introduces an inclusion-exclusion based expression for Schubert polynomials of pattern-avoiding permutations and proposes a general framework for such expressions.

Findings

Proved nonnegativity for permutations avoiding 1432 and 1423.

Partially confirmed a conjecture on principal specializations.

Established a framework for inclusion-exclusion expressions of all Schubert polynomials.

Abstract

We prove that an inclusion-exclusion inspired expression of Schubert polynomials of permutations that avoid the patterns 1432 and 1423 is nonnegative. Our theorem implies a partial affirmative answer to a recent conjecture of Yibo Gao about principal specializations of Schubert polynomials. We propose a general framework for finding inclusion-exclusion inspired expression of Schubert polynomials of all permutations.