Characterization of web permutations with the canonical matching

Characterize the web permutations of the set [n] whose associated matching M(σ) equals the canonical matching M⁽ⁿ⁾₀ = {{0,1},{2,3},…, {2n−2,2n−1}}.

Background

Web permutations are permutations obtained through the smoothing-and-switching resolution process on grid configurations. Each web permutation σ has an associated matching M(σ) on the boundary labels of the fully resolved grid configuration. The subset considered in this problem consists of those web permutations whose matching is the unique matching M⁽ⁿ⁾₀ that is simultaneously noncrossing and nonnesting.

The paper proves the Seidel-triangle enumeration conjecture for these permutations, including refined counts according to the first letter. However, the authors state that they do not know how to give an intrinsic characterization of the web permutations satisfying M(σ) = M⁽ⁿ⁾₀. The open problem is motivated by the cycle characterization of web permutations in terms of André cycles and seeks a direct description of this particular matching class.

References

Although we can prove Conjecture 1.2, we have no idea how to characterize web permutations of [n] with M(σ) = M⁽ⁿ⁾₀. In view of Theorem 1.1, we pose the following open problem. Problem 2.2. Characterize web permutations of [n] with M(σ) = M⁽ⁿ⁾₀.

— Web permutations, Seidel triangle and normalized $γ$-coefficients  (2502.01161 - Dong et al., 3 Feb 2025) in Problem 2.2, Section 2.1, p. 8