Characterize stretched-exponential asymptotics

Characterize which classes of pattern-avoiding inversion sequences or permutations exhibit stretched-exponential factors in their dominant asymptotics, and determine what structural properties govern that behavior.

Background

The paper observes that Classes 247 and 759 have estimated asymptotic forms involving stretched-exponential factors of the form μ1n3/8\mu_1^{n^{3/8}}, unlike most of the other classes studied. This behavior is compared with conjectured asymptotics for 1324-avoiding permutations.

The authors explicitly identify the determining principle behind stretched-exponential behavior as an open question. Resolving it would provide a general criterion connecting the structure of an avoidance class with the form of its asymptotic growth.

References

It is very much an open question as to what determines whether a particular class of pattern-avoiding inversion sequences or permutations have stretched exponential behaviour.

— Completing the enumeration of inversion sequences avoiding triples of relations  (2512.21943 - Britt et al., 26 Dec 2025) in Section 4, subsection “Open questions and future work”

The relaxed rule is weaker than the exact condition, and the gap between them is open. A $(V)$ pair avoiding $1324$ is a domino, so the unrestricted count is BBEP's Theorem 3.1 with growth rate $27/4$, while the strict and relaxed rules cut out subclasses whose growth rates are unknown.

— A new lower bound for the growth rate of Av(1324)  (2608.20292 - Norton, 20 Aug 2026) in Section What remains