Permuton limit for 2413-avoiding permutations

Determine the permuton limit of uniformly random permutations in the class Av(2413), whose limiting permuton is not identified by the paper.

Background

The class Av(2413) remains unresolved because the paper does not identify its limiting permuton. The authors note that rotational symmetry rules out both the anti-diagonal permuton and the diagonal permuton as possible limits. They further observe that the related class Av(2413,3142) has a random permuton limit, suggesting that the limit for Av(2413) may have a more complicated structure than the deterministic limits analyzed in the paper.

References

We conjecture that both $Av{1342}$ and $Av{1324}$ have permuton limits of $$, whereas for $Av{2413}$ the permuton limit is a bit of a mystery still.

— Permuton limits for some permutations avoiding a single pattern  (2502.19541 - Hohmeier et al., 26 Feb 2025) in Section Open Questions