Asymptotics of principal evaluations of Schubert polynomials for layered permutations

Abstract

Denote by u(n) the largest principal specialization of the Schubert polynomial: u(n) := max w∈Sn Sw(1, . . . , 1) Stanley conjectured in [Sta] that there is a limit lim n→∞ 1 n 2 log u(n), and asked for a limiting description of permutations achieving the maximum u(n). Merzon and Smirnov conjectured in [MeS] that this maximum is achieved on layered permutations. We resolve both Stanley's problems restricted to layered permutations. u(n) := max w∈Sn Υ w .

Hook formulas for skew shapes IV. Increasing tableaux and factorial Grothendieck polynomials

“…7.12. Following the approach of Stanley [S2], we conjecture that for all β ≥ 0, there is a limit In [MPP4], we computed the limit above for β = 0, when the maximum is restricted to layered (231-and 312-avoiding) permutations. It would be interesting to see if our analysis can be extended to the case of general β > 0.…”

Principal Specialization of Dual Characters of Flagged Weyl Modules

“…, 1) of the Schubert polynomial S w in terms of the reduced words of the permutation w. Fomin and Kirillov [8] placed this expression in the context of plane partitions for dominant permutations, while after two decades Billey et al [2] provided a combinatorial proof. Principal specialization of Schubert polynomials has inspired a flurry of recent interest [9,16,17,19,23,24]. A major catalyst for the current line of study into S w (1, .…”

The Sperner property for 132‐avoiding intervals in the weak order

“…See Section 2.2 for the definition of the Schubert polynomials . These polynomials are a central object of study in algebraic combinatorics and combinatorial algebraic geometry, and their principal specializations have received considerable study [9, 11, 13].…”