Permuton limits for 1342- and 1324-avoiding permutations

Establish that uniformly random permutations in the classes Av(1342) and Av(1324) converge, in the permuton sense, to the anti-diagonal permuton.

Background

The paper studies permuton limits of uniformly random permutations in several single-pattern-avoiding classes that are related by bijections to monotone-pattern-avoiding classes. The authors explain that their methods do not determine the limiting permutons for all classes avoiding a single pattern of length four. They specifically conjecture that the 1342- and 1324-avoiding classes have deterministic anti-diagonal limits, extending the collapse phenomenon established for the classes treated 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