Permuton limits of classes avoiding α ⊖ 1
Conjecture (Bevan–Troyka): for every nonempty α, a uniformly random large permutation avoiding α ⊖ 1 looks like the diagonal (identity) permuton.
Cite
@misc{cairn-permuton-limit-av-alpha-minus-one,
title = {Permuton limits of classes avoiding α ⊖ 1},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/permuton-limit-av-alpha-minus-one}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
} Also: CITATION.cff · Atom feed of results
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The problem
The question
A permuton is the scaling limit of large permutations viewed as point clouds in the unit square. For a permutation α, α ⊖ 1 is α followed by a new smallest entry (skew sum). Conjecture: for every nonempty α, the permuton limit of a uniform random permutation in Av(α ⊖ 1) is the diagonal (increasing) permuton. Related conjectures from the same session describe bases B (e.g. all patterns ending in 1, or all skew-decomposable) for which Av(B) has the increasing permuton as its limit.
What counts as progress
- Proofs for families of α.
- Large-scale simulations (Boltzmann or Markov chain samplers) for specific α with quantified distance to the diagonal, which could also reveal a counterexample.
Source. Posed by David Bevan and Justin Troyka in the open problem session of the Oberwolfach workshop Mini-Workshop: Permutation Patterns (2024), recorded in Oberwolfach Reports 6/2024, p. 282 (EMS Press, DOI 10.4171/OWR/2024/6), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.