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”