Forward stability in the 132-avoiding class

Establish that for independent permutations u,v drawn uniformly from the avoidance class Av_n(132), the expected forward stability satisfies E[FS(u,v)] = 2n − 5 + o(1) as n→∞.

Background

The authors prove that the record-set statistic is equidistributed on Av_n(132) and Av_n(231), implying identical FS distributions for these classes. Thus, the conjectured constant −5 applies equally to Av_n(132).

Verifying this would cement the link between pattern-avoidance structure and stabilization in the record-sparse regime.

References

Conjecture [Record-sparse regime] As n→∞, the following hold. (b) ($132$-avoiding.) If u,v∼ Unif{Av_n(132)}, then \E{\FS(u,v)} = 2n-5+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]