Asymptotics of principal evaluations of Schubert polynomials for layered permutations

Abstract:

Denote by $u(n)$ the largest principal specialization of the Schubert polynomial \begin{equation*} u(n) := \max _{w \in S_n} \hskip .06cm \mathfrak {S}_w(1,\ldots ,1). \end{equation*} Stanley conjectured that there is a limit \begin{equation*} \lim _{n\to \infty } \frac {1}{n^2} \hskip .06cm \log u(n), \end{equation*} and asked for a limiting description of permutations achieving the maximum $u(n)$. Merzon and Smirnov conjectured in [Eur. J. Math. 2 (2016), pp. 227–245] that this maximum is achieved on layered permutations. We resolve both of Stanley’s problems restricted to layered permutations.