Forward stability in the covexillary class

Establish that for independent permutations u,v drawn uniformly from the covexillary class CoVex_n = Av_n(3412), there exists a constant c_cov ∈ [6, 9] such that E[FS(u,v)] = 2n − c_cov − o(1) as n→∞.

Background

Covexillary permutations play a prominent role in the local geometry of Schubert varieties (e.g., through positive combinatorial formulas for singularity invariants). The conjecture posits a constant-order gap below 2n in the record-sparse regime.

Determining c_cov would give a sharp quantitative stabilization law for this important class.

References

Conjecture [Record-sparse regime] As n→∞, the following hold. (e) (Covexillary.) If u,v∼ Unif{CoVex_n}, then there exists a constant c_{\mathrm{cov}∈[6,9] such that \E{\FS(u,v)} = 2n - c_{\mathrm{cov} - o(1).

— 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]