2 problems

Fix $k\in\mathbb{Z}_{>0}$, let $\zeta$ be a $k$-th root of unity, and let $\Phi_w^k$ denote the principal specialization $\mathfrak{S}_w(1,q,q^2,\ldots)$ at $q=\zeta$. Define … and…

Doubly layered maximizer conjecture. For every $n$, all permutations reaching the maximum $\widetilde{u}(n)$ are doubly layered permutations. Thus, there is a limit