34 problems

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…

The Stanley-Wilf and subsequence-constant conjecture. Both limit superiors are actual limits, and

Tóth's conjecture. The quantity

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…

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 $(\tau_n)_n$ 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…

The threshold interpolation conjecture. If $n^{1-1/s}\ll m$ and $\binom{n}{2}-m\gg n^{1-1/s'}$, 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 $n\to\infty$,

Tóth's conjecture. Almost surely, $\pi_\beta$ has only finite cycles when $d=2$, for every $\beta>0$, whereas when $d\geq3$ it has infinite cycles for every $\beta>\beta_c$.

For integers $k\leq n$, let $U_k(n)$ be the minimum length of a sequence of independent lazy transpositions whose product maps the elements $1,\dots,k$ uniformly to the…

Let $F_{\rm \max}$ be the maximal value function whose derivative appears as the limit shape in the paper's limit-shape corollary, and let $F_{\rm \max}'$ denote that limit shape. Co…

Let $\Omega$ be the open diamond … and let $\Phi$ be the limiting variational function for a homogeneous Poisson point process on $\Omega$; its derivative $\Phi'$ is the correspond…

Let $n\geq 3$. A smooth Mallows process is a process … for which the associated graph group $\langle{\mathcal{G}}_{\mathcal{M}}\rangle$ is defined, and let $\mathcal{T}$ denote the…

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…

Non-central moment conjecture. For all $r\ge 1$, if $n$ is sufficiently large in terms of $m$, then

Central moment conjecture. For all $r\ge 1$, if $n$ is sufficiently large in terms of $m$, then

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…

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

Let a random walk of length $2n+1$ be generated from a uniform random permutation, and let $2k$ be the number of its edges lying above the origin. Write $\alpha_{2n,2k}$ for the co…

Let $\mathcal{S}_k$ denote the permutations of $[k]$. Fix $\varepsilon>0$ and set $n=(1+\varepsilon)k^2/4$. For $\pi\in\mathcal{S}_k$, let $\sigma$ be a uniformly random permutatio…

Let $k$ be a positive integer. Let $\pi$ be a permutation of length $k$, let $n=1000k^2$, and let $\sigma$ be a uniformly random permutation of length $n$. Containment probability…

Let $k$ be a positive integer and let $\mathcal{S}_n$ be the set of permutations of $[n]$. A permutation is $k$-universal if it contains every pattern in $\mathcal{S}_k$. For fixed…