17 problems

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 $\nu$.…

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…

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

Bernoulli-number conjecture. The coefficients satisfy

Hypergeometric identity conjecture. The following identity should hold:

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…

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

Let $n\geq 1$, and let $\tau$, $\mu$, and $\nu$ be partitions of $n$. For the coefficients $h_{\mu,\nu}^{\tau}(\beta)$ defined by the Jack-polynomial generating series, let the sum…

The $b$-conjecture. The series $\Psi(\mathbf{x},\mathbf{y},\mathbf{z};t,1+b)$ has coefficients that are polynomials in $b$ with non-negative integer coefficients. The constant term…

Let $\pi$ be a partition, let $\lambda$ be a Young diagram, and let $\alpha>0$. Denote by $\operatorname{Ch}_\pi^{(\alpha)}$ the Jack character and by…

Let $\lambda$ be a Young diagram, let $\pi$ be a partition, let $\alpha>0$, and let $M$ range over non-oriented maps with face-type $\pi$. Write $V_\bullet(G)$ and $V_\circ(G)$ for…