Asymptotics of principal evaluations of Schubert polynomials for layered permutations

TL;DR

This paper investigates the asymptotic behavior of the maximum principal evaluations of Schubert polynomials for layered permutations, confirming conjectures about their growth rate and the permutations that achieve this maximum.

Contribution

It resolves Stanley's conjecture on the limit of the scaled logarithm of maximum Schubert polynomial evaluations for layered permutations.

Findings

Established the limit of (1/n^2) log u(n) for layered permutations.

Confirmed that layered permutations achieve the maximum evaluations.

Provided a limiting description of permutations attaining the maximum.

Abstract

Denote by $u(n)$ the largest principal specialization of the Schubert polynomial: $ u(n) := \max_{w \in S_n} \mathfrak{S}_w(1,\ldots,1) $ Stanley conjectured in [arXiv:1704.00851] that there is a limit $\lim_{n\to \infty} \, \frac{1}{n^2} \log u(n), $ and asked for a limiting description of permutations achieving the maximum $u(n)$. Merzon and Smirnov conjectured in [arXiv:1410.6857] that this maximum is achieved on layered permutations. We resolve both Stanley's problems restricted to layered permutations.