37 problems

Let $b=\frac{2}{\beta}-1$. The hypermap series enumerates bipartite maps with independently specified profiles of the two types of vertices, and its coefficients are denoted by…

Fully simple maps conjecture. The generating functions for the weighted numbers of fully simple maps satisfy topological recursion.

Bessis–Itzykson–Zuber conjecture. The logarithm of the normalized partition function has an asymptotic expansion

Constellation enumeration conjecture. The number of $m$-constellations having $n$ black faces is

Let $g\geq 0$ and $p\geq 3$ be fixed integers, and let $\boldsymbol{\alpha}=(\alpha_1,\alpha_2,\ldots,\alpha_p)$ satisfy $\alpha_k>0$ and $\sum_{k=1}^p\alpha_k=1$. Define … For…

Ercolani–Lega–Tippings conjecture. For every $g\geq 2$,

For all $n,f\geq 1$ and partitions $\nu\vdash 2n$, let $c_{\mu,\nu,n}(b)$ denote the coefficient associated with maps having $n$ edges and vertex and face profiles $\mu$ and $\nu$.…

Heuristic asymptotics. The number of triangulations satisfies

Few-monochromatic-edges conjecture. For fixed $i,j\geq 0$, there exists a rational function $Q_{i,j}(g)$ in $g$ such that

Many-monochromatic-edges conjecture. For fixed $i,j\geq 0$, there exists a rational function $R_{i,j}(g)$ in $g$ such that

The Macdonald generalization of the Matchings-Jack conjecture. For any positive integer $n$ and partitions $\pi,\mu,\nu$ of $n$,

The b-conjecture. The coefficients $h^\pi_{\mu,\nu}$ are polynomials in $b$ with non-negative integer coefficients. The paper explicitly says both Jack conjectures remain open.

The b-conjecture. The coefficients $c^\pi_{\mu,\nu}$ are polynomials in $b$ with non-negative integer coefficients. The paper states that the conjecture remains open, although its…

Ercolani–Lega–Tippings' conjecture. When $\nu=2$, the summands in the formula for the number of $2\nu$-valent maps with $j$ vertices and two legs embedded in a surface of genus…

Hanlon's conjecture. There exists a statistic $w$ on the orientable maps in $\operatorname{\mathcal{ID}^O}_n(\lambda)$ such that

Injective map expansion conjecture. There exists a statistic of non-orientability $\vartheta$ on $\operatorname{\mathcal{ID}}_n(\lambda)$ such that

Hypergeometric identity conjecture. The following identity should hold:

Genus expansion conjecture. In the large-$n$ limit, $\mathcal G_N^{(n)}$ should have, in some sense, an asymptotic expansion in inverse powers of $n^2$ of the form

Generalized fully simple maps conjecture. The generating functions of generalized fully simple maps satisfy topological recursion.

$b$-conjecture. For all partitions $bb,bc,bd\vdash n$, $h^{\lambda}_{\mu,\nu}(b)\in\mathbb{N}[b]$, and there is a statistic $\operatorname{stat}$ on these maps such that

Quadrangulation conjecture. There is an explicit bijection preserving genus and number of edges between rooted quadrangulations and rooted maps with certain decorations that proves…

Let $\mathcal{C}_{n}(\\{T_{2k+1},B_{2k+1}\\}_{k\geqslant 1})$ be the set of connected chord diagrams of size $n$ avoiding all odd top and bottom cycles. The cubic-map enumeration c…

Unified bijection conjecture. A unified bijection for essentially irreducible toroidal $d$-angulations should exist.

Let $n$ be a positive integer and let $\pi,\sigma,\lambda\vdash n$ be partitions. Let $M^\lambda_{\pi,\sigma}$ be the set of rooted maps indexed by these partitions, and call a map…

Fully simple amplitude conjecture. For ordinary maps or maps with loops,