Forward stability in the fireworks class

Establish that for independent permutations u,v drawn uniformly from the fireworks class 𝔽_n, the expected forward stability satisfies E[FS(u,v)] = 2n βˆ’ 2n/(log n βˆ’ log log n) + o(n/log n) as nβ†’βˆž.

Background

Fireworks permutations are defined via increasing initial elements of maximal consecutive decreasing runs. The paper’s computations indicate markedly sparse record structures leading to stabilization very close to 2n, with a slowly varying subtractive term of order n/log n.

This conjecture pinpoints the asymptotic correction for this class within the record-sparse regime.

References

Conjecture [Record-sparse regime] As nβ†’βˆž, the following hold. (c) (Fireworks.) If u,v∼ Unif{\mathcal F_n}, then \E{\FS(u,v)} = 2n-\frac{2n}{\log n-\log\log n} +o!\left(\frac{n}{\log n}\right).

— 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]