27 problems

A Baxter permutation of size $n$ is a permutation avoiding the two patterns specified in the source, and bipolar-oriented planar maps with general face degrees correspond to Baxter…

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…

For every integer $n\geq 2$, let $M_n$ be a uniformly distributed random rooted $2p$-angulation over ${\cal M}^p_n$, let ${\bf m}_n$ be its vertex set, and let $d_n$ be its graph d…

Type III branching-process conjecture. For type III triangulations, one can construct $\zeta_3$ and $\bar{\zeta}_3$ so that the analogue of the branching-process theorem holds. The…

A planar map is a planar graph together with its embedding, and an automorphism is a symmetry preserving the relevant map structure. Tutte's asymmetry conjecture. Almost all planar…

A quartic Eulerian orientation without alternating vertices is a quartic Eulerian orientation belonging to the subclass obtained by excluding alternating vertices. Bousquet-Mélou a…

Greedy $m$-Tamari intervals are intervals in the greedy $m$-Tamari poset, and planar $(m+1)$-constellations are planar maps with the relevant face-degree profile. Bousquet-Mélou an…

Let $\mathbb{X}\subset\mathbb{S}^2$ be the embedded random surface, and let $\mathbb{X}'$ be an independent copy of $\mathbb{X}$. A neighborhood of $\mathbb{X}$ means a neighborhoo…

Let $\mathbb{X}\subset\mathbb{S}^2$ be the embedded random surface, and let $\mathscr{G}$ be the graph whose vertices are the faces of $\mathbb{X}$, with an edge between two vertic…

Let $a\from (2,5/2)$, let $\bigl(f_i\bigr)_{i\geq 1}$ be the sampled marked vertices or faces, let $d^\dagger_\mathrm{gr}$ denote the graph distance, and let…

Let $a\from (3/2,5/2)$, let $\bigl(f_i\bigr)_{i\geq 1}$ be the sampled marked vertices or faces, let $d^\dagger_\mathrm{fpp}$ denote the first-passage percolation distance, and let…

Gasket convergence conjecture. For $n=2$, the gasket should converge in some sense to the $\mathrm{CLE}_4$ ensemble.

A planar linear normal $\lambda$-term is a linear normal $\lambda$-term whose syntactic diagram is planar; call it 3-connected when its syntactic diagram is 3-connected. For…

Let $\mathrm{M}_n(\mathrm{f}_n,\mathrm{g}_n)$ be random maps with face parameters satisfying $1\ll \mathrm{f}_n+2\mathrm{g}_n\ll n$. In the regime where…

A rooted series-parallel map is the rooted planar-map model studied in the cited work; the source notes that these maps are in bijection with separable permutations. The critical…

The infinite Brownian disk with boundary length $\mathcal{L}_1$ is a random metric space homeomorphic to the complement of the open unit disk in the plane. Shortest-loop conjecture…

Let $t$ be the generating-function variable, let $k$ be the approximation parameter, and let $\bar\mu_k$ denote the estimated growth rate associated with the algebraic generating s…

Let $\mathscr{K}_n$ and $\mathcal{K}_n$ denote, respectively, the classes of knot shadows and knot diagrams with $n$ crossings, and write $k_n=|\mathscr{K}_n|$ and…

For $n\in\mathbb N$ and $q\geq 0$, let $(M,e_0,S)$ be a rooted critical FK planar map with parameter $q$ and size $n$, and let $T$ be the spanning tree obtained from the FK edge se…

Let $M$ be a planar map with $n$ edges, embedded into $\mathbb C$ by circle packing or Riemann uniformization, and let $T$ be a spanning tree of $M$. Let the Peano curve of $T$ be…

Let $G$ be a graph, and let an activity be a rule assigning active edges to spanning trees and describing the Tutte polynomial. Suppose it satisfies Crapo's property, namely that t…

Stable-looptree universality conjecture. The family of stable looptrees should appear generically in the study of interfaces in statistical mechanical models on planar maps.

Let $M_n$ be a uniform planar map with $n$ edges and let $Q_n$ be a uniform planar quadrangulation with $n$ faces. Equip both maps with their graph distances, and let the Ambjørn–B…

Let $F=F(z,u)$ be the generating series for the spanning forests considered in the 4-valent case, with $z$ marking the relevant map parameter and $u$ the forest parameter. Write…

Let $Q_\infty$ be the uniform infinite planar quadrangulation (UIPQ), rooted at $\rho$, and let $(X_k)_{k\geq 0}$ be simple random walk on $Q_\infty$ started at $\rho$. Write…