26 problems

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…

Brownian separable permuton construction conjecture. Brownian separable permutons can be constructed using $\text{SLE}_4/\text{CLE}_4$ and critical Liouville quantum gravity with…

Permuton-limit conjecture. Both $\textbf{Av}_n(1342)$ and $\textbf{Av}_n(1324)$ have permuton limits equal to $\mu_J$; that is, uniformly random permutations from each class conver…

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…

Uniform bounds conjecture. There exist constants $0<c<C$, depending on all the data, such that $$

For fixed $p,q\in(0,1)$, let $\boldsymbol{w}^{(q)}(n)\in S_n$ be the random permutation from the $q$-reduced pipe-dream model, and let $H^{(q)}$ be its height function. Let…

Universal height lower-bound conjecture. For any permuton $cmu$ and $cvarepsilon>0$,

Injectivity conjecture. Then

Skew Brownian LIS exponent conjecture. There exists a function $\ell:[-1,1]\times(0,1)\to[1/2,1)$ such that, with probability tending to $1$ as $n\to\infty$,

Mallows-limit conjecture. The permutons $\bm{\mu}_{-1,q}$ are the Mallows permutons.

For $(\rho,q)\in(-1,1)\times[0,1]$, let $\mu_{\rho,q}$ denote the corresponding skew Brownian permuton, and let $\widetilde{\operatorname{occ}}(21,\mu_{\rho,q})$ be its normalized…

Let $\bm \mu_{\rho,q}$ be a skew Brownian permuton with parameters $(\rho,q)\in(-1,1]\times[0,1]$, and define … Here $21$ is the inversion pattern of size two. Inversion monotonici…

Let $\bm \mu_{\rho,q}$ be a skew Brownian permuton with parameters $(\rho,q)\in(-1,1)\times(0,1)$. For $k\in\mathbb{Z}_{>0}$, let $\pi$ be a permutation pattern of size $k$, and le…

Semi-Baxter and strong-Baxter universality conjecture. The permuton limit of uniform semi-Baxter or strong-Baxter permutations is the skew Brownian permuton. In particular, for str…

Skew Brownian permuton well-definedness conjecture. The skew Brownian permuton is well-defined for every parameter pair

Biased-separable representation conjecture. For every $p\in[0,1]$,

Corner permuton limit conjecture. If $k,n\to\infty$ with $k/n\to\alpha\in\mathbb{R}_{>0}$, then

Phase-transition conjecture. If $\bm\sigma_n^k$ is uniform in $ASq(n,k)$, then the following additional regimes hold:

Lyndon-permutation dimension conjecture. The feasible region $clP_k$ contains an open ball of dimension $|\mathcal{L}_k|$.

Skew Brownian permuton universality conjecture. The skew Brownian permuton describes a new universal limiting object that includes both the Baxter permuton and the biased Brownian…

Universal scalable convergence conjecture. If $\Gamma$ is any permuton, then every $\Gamma$-random sequence is scalably convergent.

Tieredness conjecture. If $\Gamma$ is a permuton for which every $\Gamma$-random sequence is scalably convergent, then $\Gamma$ is tiered.

Archimedean process convergence conjecture. $\mathbf{RSN}^n$ converges in probability, as a permutation process, to the Archimedean process.

Archimedean path conjecture. For every $t$, the random permutation $\mathbf{RSN}^n_{\lfloor tN\rfloor}$ converges to the deterministic permuton $(\mathcal{A}(0),\mathcal{A}(t))$; e…