Enumeration and asymptotics of higher stabilization levels

Determine an efficient method for computing the cardinalities of the inversion-sequence sets I_{n,k} for fixed k≥2, or determine their asymptotic behavior.

Background

The sets I_{n,k} consist of inversion sequences whose k-th Θ-image is fixed. The paper gives Catalan and semi-Baxter descriptions for the first two stabilization levels, but does not obtain a comparable enumeration for fixed levels k≥2.

The unresolved task is both algorithmic and asymptotic: one seeks either an efficient procedure for calculating |I_{n,k}| at each fixed higher level or a description of its growth as n tends to infinity.

References

For fixed $k\geq2$, find a way to efficiently compute the numbers $|I_{n,k}|$, or determine their asymptotic behavior. While the cases $k=0$ and $k=1$ admit Catalan and semi-Baxter descriptions, respectively, no comparable enumeration is presently known for higher stabilization levels.

— Iterating the Lehmer code on inversion sequences: Catalan fixed points and finite stabilization  (2608.24476 - Allagan et al., 25 Aug 2026) in Problem 2, labeled Problem higher_generating, Section 5 (Conclusion and open problems)