Structural characterization of higher stabilization levels

Characterize the structural properties of the permutation sets S_{n,k} and inversion-sequence sets I_{n,k} for each fixed k≥2.

Background

The sets S_{n,k} contain permutations that reach a fixed point after at most k iterations of Θ, while I_{n,k} is the analogous class of inversion sequences. The paper provides a pattern-avoidance characterization for k=1, but shows that the union of the k=2 permutation sets is not closed under classical pattern containment.

Consequently, the higher stabilization levels do not appear to admit a straightforward description within classical permutation-pattern theory. The authors explicitly identify the exact behavior of these levels as largely unexplored, leaving open the task of finding structural characterizations, potentially using methods beyond traditional permutation classes.