Forward stability in the vexillary class

Establish that for independent permutations u,v drawn uniformly from the vexillary class Vex_n = Av_n(2143), the expected forward stability satisfies E[FS(u,v)] = 2n − O(√n) as n→∞.

Background

Vexillary permutations have determinantal and flagged-Schur descriptions for their Schubert polynomials. The authors’ experiments suggest a record-sparse regime with a sublinear √n correction below 2n.

Proving this would clarify stabilization in a class central to algebraic combinatorics and Schubert calculus.

References

Conjecture [Record-sparse regime] As n→∞, the following hold. (d) (Vexillary.) If u,v∼ Unif{Vex_n}, then \E{\FS(u,v)} = 2n - O(\sqrt{n}).

— The record statistic and forward stability of Schubert products  (2604.02964 - Hardt et al., 3 Apr 2026) in Section 7 (Conjectures), Conjecture [Record-sparse regime]