Upper bounds of Schubert polynomials

Let w be a permutation of{1,2,...,n},and let D(w)be the Rothe diagram of w.The Schubert polynomial■w_(x)can be realized as the dual character of the flagged Weyl module associated with D(w).This implies the following coefficient-wise inequality:Min_(x)≤■_(w)(x)≤Max_(w)xwhere both Min_(w)(x)and Max_(w)(x)are polynomials determined by D(w).Fink et al.(2018)found that■w_(x)equals the lower bound Min_(w)(x)if and only if w avoids twelve permutation patterns.In this paper,we show that■w_(x)reaches the upper bound Max_(w)(x)if and only if w avoids two permutation patterns 1432 and 1423.Similarly,for any given compositionα∈Z^(n)≥0,one can define a lower bound Min_(α)(x)and an upper bound Max_(α)(x)for the key polynomialκ_(α)(x).Hodges and Yong(2020)established thatκ_(α)(x)equals Min_(α)(x)if and only ifαavoids five composition patterns.We show thatκ_(α)(x)equals Max_(α)(x)if and only ifαavoids a single composition pattern(0,2).As an application,we obtain that whenαavoids(0,2),the key polynomialκ_(α)(x)is Lorentzian,partially verifying a conjecture of Huh et al.(2019).