8 problems

Scaling-limit conjecture. As $n\to\infty$, the following limits hold. If $0\leq\alpha_+<\tilde\alpha_+$ and $0\leq\alpha_-<\tilde\alpha_-$ are fixed, then

Chen's conjecture. For any fixed $n$, the sequence $\ell_{n,1},\ell_{n,2},\ldots,\ell_{n,n}$ is log-concave. Equivalently, the distribution of $\ell_n(\sigma)$ for a uniformly chos…

Let $(\tau_n)_n$ be a sequence of conjugation-invariant random permutations, where $\tau_n$ has size $n$ and random cycle type $t^{(n)}$. Suppose that … in probability. Conjecture…

Let $L_n^{\varoast}$ denote the relevant longest monotone subsequence length for $\varoast\in\{\boxslash,\boxbslash,\boxdot\}$, let $\gamma(\varoast)$ and $\delta(\varoast)$ be the…

Let $l_N^{\boxbslash}$ and $l_N^{\boxslash}$ denote the lengths of the two symmetric longest increasing subsequences considered in the source, and let…

Let $l_N^\square$ be the length of the longest increasing subsequence in the finite $N$ model, and let $E_2^{\rm soft}(0;(t,\infty))$ denote the soft-edge limiting distribution. Se…

Let $n$ be a positive integer. For $1\leq k\leq n$, let $I_{n,k}$ be the set of involutions in the symmetric group $\mathfrak{S}_n$ whose longest increasing subsequence has length…

Fluctuation conjecture. If $0<\alpha<\frac12$, then