5 problems

Let $\sigma_n$ be a uniformly random permutation in $S_n$, and let $L(\sigma_n)$ denote the length of its longest increasing subsequence. Hammersley's conjecture. The limit … exist…

Let $\sigma^{\mathrm{Unif}}_n$ be a uniformly random permutation of $\{1,\dots,n\}$, and let $\operatorname{LIS}(\sigma)$ denote the length of its longest increasing subsequence. U…

Let $\operatorname{Maj}(n,q)$ be the random permutation produced by the major-index sampler with parameter $q$, and let $\operatorname{LIS}(\operatorname{Maj}(n,q))$ denote its lon…

Let $(\sigma_n)_{n\geq 1}$ be conjugacy-invariant random permutations of size $n$. Suppose that … and … Writing $\lambda(\sigma_n)$ for the partition associated with $\sigma_n$ and…

Gaussian fluctuations conjecture. For every $s\in\mathbb{R}$,