Enumeration of permutations by stack-sorting depth

Determine the number of permutations whose stack-sorting depth is at most a given integer t for general t, extending the known enumerations for small depths.

Background

The paper defines the stack-sorting depth of a permutation as the minimum number of applications of West’s stack-sorting map required to sort it. Enumeration is known for depth at most 1, via the Catalan numbers, and for depth at most 2, via an explicit formula. Results for depth at most 3 include a polynomial-time counting algorithm, but the general enumeration problem remains unresolved, particularly for depths t greater than or equal to 4.

References

The problem of enumerating permutations \pi with \langle \pi\rangle \le t is open for general t.

— Motzkin Numbers Count 2-Stack-Sortable Permutations Ending in Their Least Entry  (2610.02756 - Inagaki et al., 2 Oct 2026) in Section 1, Introduction