Enumeration of 321-avoiding non-crossing and non-nesting permutations

Enumerate exactly the non-nesting and non-crossing permutations on the multiset {1,1,2,2,...,n,n} that avoid the pattern 321, thereby determining their respective counting sequences beyond the known bounds between the Catalan number C_n and C_n^2.

Background

The paper develops generating functions for non-nesting and non-crossing permutations avoiding the pattern 231, resolving one of the previously posed single-pattern enumeration questions for both classes. It also treats several cases involving the repeated-element pattern 122 and additional patterns.

The authors identify avoidance of 321 as a principal unresolved direction. They state that the enumeration problem remains open for both non-nesting and non-crossing permutations, while noting that the corresponding counts are bounded between the Catalan number C_n and its square C_n2. Exact enumeration would determine the growth and structure of these two pattern-avoidance classes.

References

In particular, it is still open to enumerate non-nesting and non-crossing permutations that avoid the pattern 321.

— Pattern avoidance in non-crossing and non-nesting permutations  (2502.13309 - Archer et al., 18 Feb 2025) in Section 5, Open Questions

In particular, it is still open to enumerate non-nesting and non-crossing permutations that avoid the pattern 321. It is clear that both pn (321) and pn (321) are bounded between Cn and C2, but enumerating them exactly seems to challenging.

— Pattern avoidance in non-crossing and non-nesting permutations  (2502.13309 - Archer et al., 18 Feb 2025) in Section 5, Open Questions