Forward stability in the smooth class

Establish that for independent permutations u,v drawn uniformly from the smooth class Sm_n = Av_n(3412,4231), there exists a constant c_sm ∈ [1.35, 1.40] such that E[FS(u,v)] = c_sm n + o(n) as n→∞.

Background

Smooth permutations correspond to Schubert varieties that are smooth; they avoid the patterns 3412 and 4231. The authors’ experiments suggest an intermediate regime in which the linear growth rate lies strictly between 1 and 2.

Proving this would quantify stabilization in a key geometric class and connect avoidance structure with linear-rate stabilization.

References

Conjecture [Intermediate record regime] As n→∞, the following hold. (a) (Smooth.) If u,v∼ Unif{Sm_n}, then there exists a constant c_{\mathrm{sm}∈[1.35,1.40] such that \E{\FS(u,v)} = c_{\mathrm{sm} n + o(n).

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