Upper bounds of Schubert polynomials

Let w be a permutation of {1, 2, …, n}, and let D(w) the Rothe diagram w. The Schubert polynomial ${\mathfrak{S}_w}\left(x \right)$ can realized as dual character flagged Weyl module associated with D(w). This implies following coefficient-wise inequality: $${\rm{Mi}}{{\rm{n}}_w}\left(x \right) \le {\mathfrak{S}_w}\left(x {\rm{Ma}}{{\rm{x}}_w}\left(x \right),$$ where both Minw(x) Maxw(x) are polynomials determined by Fink et al. (2018) found that equals lower bound if only avoids twelve patterns. In this paper, we show reaches upper two patterns 1432 1423. Similarly, for any given composition α ∈ ℤ ≽0 n , one define Minα(x) an Maxα(x) key κα(x). Hodges Yong (2020) established κα(x) five We single pattern (0, 2). As application, obtain when 2), is Lorentzian, partially verifying conjecture Huh (2019).