Forward stability in the 321-avoiding class

Establish that for independent permutations u,v drawn uniformly from the avoidance class Av_n(321), there exists a constant c_321 ∈ [0.92, 0.94] such that E[FS(u,v)] = n + c_321√n + O(1) as n→∞.

Background

Pattern-avoidance classes often exhibit distinctive record statistics that drive forward stability via the Hardt–Wallach max formula. Numerical experiments in the paper suggest a √n-order correction to n for the 321-avoiding class, with a constant in a narrow interval.

This places Av_n(321) within the record-dense regime while quantifying the gap above n.

References

Conjecture [Record-dense regime] As n→∞, the following hold. (c) ($321$-avoiding.) If u,v∼ Unif{Av_n(321)}, then there exists a constant c_{321}∈[0.92,0.94] such that \E{\FS(u,v)} = n+c_{321}\sqrt{n}+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-dense regime]