Characterization of web permutations with the minimal noncrossing-nonnesting matching

Characterize the web permutations of [n] whose associated matching M() equals the matching M_0^{(n)} = \{\{0,1\},\{2,3\},\ldots,\{2n-2,2n-1\}\}.

Background

For a web permutation in the symmetric group S_n, the paper associates a matching M() on the boundary labels {0,1,\ldots,2n-1} of its fully resolved grid configuration. The matching M_0{(n)} is the unique matching that is simultaneously noncrossing and nonnesting. The subset of web permutations satisfying M()=M_0{(n)} is denoted by \widetilde{\mathrm{Web}}_n and is central to the paper's refinement of the Seidel triangle.

References

Although we can prove Conjecture 1.2, we have no idea how to characterize web permutations of [n] with M(\sigma) = 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(\sigma) = M{(n)}_0.

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