8 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

A layered permutation is a permutation containing neither $231$ nor $312$ as a pattern. Let $u$ be a layered permutation with layers of sizes $(ell1,ldots,ellk)$, and let a sequen…

Packing-complexity conjecture. Under these hypotheses, there exists $K$ such that

Fixed-block approximation conjecture. If $2^\ell\geq 1+k$, then there exists a sequence $(\tau_n)_{n\in\mathbb{N}}$ of layered permutations with $b$ blocks such that…

Generalized Myers conjecture. For every $k,\ell\geq 2$,

Let $\mathrm{Av}_\sigma(n)$ denote the number of permutations of length $n$ avoiding a pattern $\sigma$. Let $\sigma$ and $\tau$ be patterns of the same length, with $\tau$ layered…

A pattern is layered if it is a direct sum of decreasing permutations. For each positive integer $k$, consider all patterns of length $k$ and, for sufficiently large $n$, the numbe…