Conjecture: L_{n,2} ∼ 3√n for the wreath product action S_2^n ⋊ S_n

Establish that for k = 2, with σ uniformly random in S_2^n ⋊ S_n acting on {1,2,...,2n}, the length L_{n,2} of the longest increasing subsequence satisfies L_{n,2} ∼ 3√n as n → ∞.

Background

The authors compare two actions involving B_n (the hyperoctahedral group): the signed-permutation action, where known results give L_n(σ) ≈ 2√(2n), and the wreath product action, where their main theorem does not apply when k=2.

They conjecture a different asymptotic constant for the wreath product action at k=2, reflecting that different group actions can lead to distinct longest-increasing-subsequence behavior.