92 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…

Brownian-sheet persistence conjecture. The probability that $\pi$ is below $\tau$ in the strong Bruhat order satisfies

Let $G$ be a Cayley graph, let $(\psi_e)_{e\in E(G)}$ be independent unit-rate Poisson processes on the edges, and let $\beta>0$. At time $\beta$, swapping the particles across eac…

Let $\Omega$ be the infinite-dimensional simplex of decreasing nonnegative sequences with sum $1$, and let $M$ be the transition kernel of the continuous coagulation-fragmentation…

Tóth's macroscopic-cycle conjecture. For $d\geq3$ and $\beta>\beta_c$, macroscopic cycles form: roughly $\alpha L^d$ particles belong to macroscopic cycles and roughly…

Let $\rho$ be the uniform distribution on the diamond … Let $\Phi'$ denote the limit shape of the Young diagram associated with the corresponding locally uniform random permutation…

Fix parameters $\rho$ and $q$ in the range for which the skew Brownian permuton construction gives corresponding parameters $\gamma$ and $\theta(q)$ through the relations stated in…

Baik–Deift–Johansson conjecture. The limit distribution of the $x_i$'s exists as $n\to\infty$ and coincides with the limit distribution of the $y_i$'s.

Stanley's limiting-distribution conjecture. The limit

Let $w$ be a uniformly random permutation in the symmetric group ${\cal S}_n$, and let $\operatorname{as}_n(w)$ denote the length of its longest alternating subsequence. If…

Tóth's conjecture. The quantity

Eberhard's conjectures. The limit inferior of $n^2P_2(n)$ is finite; $n^2P_2(n)$ converges along subsequences of the arithmetic form described above; and $n^2P_2(n)$ is bounded by…

Convergence-to-the-Schubert-permuton conjecture. The random permutations distributed according to this measure converge, as $n\to\infty$, to a deterministic permuton $\mu$ on…

Conjecture. A similar result to Theorem $$ will hold for the descents of top-to-random.

Fyodorov–Muirhead conjecture. The expected diameter of the cycle satisfies

Uniform separable scaling conjecture. There is a deterministic constant $c>0$ such that

RPCD worst-case conjecture. The upper bound above, proved for the specified permutation-invariant class of quadratic Hessians, extends to all positive-definite quadratic functions.…

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…

Universality conjecture. The Brownian separable $d$-permuton should be a universal limit describing other natural classes of $d$-dimensional permutations, much like the Brownian se…

Let $\nu=\mu^R$ be the permuton described above, and let $\alpha>0$. Consider uniform permutations of size $m=n+k$ with exactly $n$ left-to-right minima and $k$ elements which are…

Fix $p\in(0,1)$, and for each $n\geq 2$ let $\mathsf{w}^{(n)}=(s_{n-1})(s_{n-2}s_{n-1})\cdots(s_1s_2\cdots s_{n-1})$ be the staircase reduced word, and let $v^{(n)}$ be the random…

Let $p\in[0,1)$ and let $\boldsymbol{w}=\boldsymbol{w}(n)\in S_n$ be the random permutation in the non-reduced model. Write $\operatorname{inv}(\boldsymbol{w})$ for its number of i…

Let $p otin[0,1)$ and let $boldsymbol{w}^{(1)}=boldsymbol{w}^{(1)}(n)in Sn$ be the random permutation in the $q=1$ model. Write $operatorname{inv}(boldsymbol{w}^{(1)})$ for its num…

Let the underlying geometry be modelled by a degree-two configuration model with rewiring, and let ISRW denote the associated random walk. Sublinear-mixing conjecture. ISRW should…